{"id":74,"date":"2019-08-20T17:02:04","date_gmt":"2019-08-20T21:02:04","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/fitting-linear-models-to-data\/"},"modified":"2022-06-01T10:39:25","modified_gmt":"2022-06-01T14:39:25","slug":"fitting-linear-models-to-data","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/fitting-linear-models-to-data\/","title":{"raw":"Fitting Linear Models to Data","rendered":"Fitting Linear Models to Data"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section you will:\n<ul>\n \t<li>Draw and interpret scatter diagrams.<\/li>\n \t<li>Use a graphing utility to find the line of best fit.<\/li>\n \t<li>Distinguish between linear and nonlinear relations.<\/li>\n \t<li>Fit a regression line to a set of data and use the linear model to make predictions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id2164914\">A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.<\/p>\n\n<div id=\"fs-id2188781\" class=\"bc-section section\">\n<h3>Drawing and Interpreting Scatter Plots<\/h3>\n<p id=\"fs-id1618160\">A <span class=\"no-emphasis\">scatter plot<\/span> is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a <span class=\"no-emphasis\">linear model<\/span>, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions. <a class=\"autogenerated-content\" href=\"#Figure_04_03_001\">(Figure)<\/a> shows a sample scatter plot.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/d85e87d6a305be35528f5ab89b201056fdbe8d53\/CNX_Precalc_Figure_02_04_001.jpg\" alt=\"Scatter plot, titled 'Final Exam Score VS Age'. The x-axis is the age, and the y-axis is the final exam score. The range of ages are between 20s - 50s, and the range for scores are between upper 50s and 90s.\" width=\"487\" height=\"337\"> <strong>Figure 1. <\/strong>A scatter plot of age and final exam score variables[\/caption]\n<p id=\"fs-id1621860\">Notice this scatter plot does <em>not<\/em> indicate a <span class=\"no-emphasis\">linear relationship<\/span>. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.<\/p>\n\n<div id=\"Example_04_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1802821\">\n<div id=\"fs-id1798376\">\n<h3>Using a Scatter Plot to Investigate Cricket Chirps<\/h3>\n<p id=\"fs-id1797122\"><a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a> shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit[footnote]Selected data from <a href=\"http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/\">http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/<\/a>. Retrieved Aug 3, 2010[\/footnote]. Plot this data, and determine whether the data appears to be linearly related.<\/p>\n\n<table id=\"Table_04_03_01\" summary=\"Table with two rows and ten columns. The first column is labeled: \u201cChirps\u201d and the second is labeled: \u201cTemperature\u201d. The values for chirps are: 44, 35, 20.4, 33, 31, 35, 18.5, 37, 26. The values for Temperature are: 80.5, 70.5, 57, 66, 68, 72, 52, 73.5, 53.\"><caption>Cricket Chirps vs Air Temperature<\/caption>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2064463\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2064463\"]Plotting this data, as depicted in <a class=\"autogenerated-content\" href=\"#Figure_04_03_002\">(Figure)<\/a> suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/0bd821366e09c7df8ba83f57a8b07d4aa8fb15aa\/CNX_Precalc_Figure_02_04_002.jpg\" alt=\"Scatter plot, titled 'Cricket Chirps vs. Air Temperature'. The x-axis is the Cricket Chirps in 15 Seconds, and the y-axis is the Temperature (F). The line regression is generally positive.\" width=\"487\" height=\"386\"> <strong>Figure 2.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2672477\" class=\"bc-section section\">\n<h3>Finding the Line of Best Fit<\/h3>\n<p id=\"fs-id1798587\">Once we recognize a need for a linear function to model that data, the natural follow-up question is \u201cwhat is that linear function?\u201d One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the <em>y<\/em>-intercept. We can approximate the slope of the line by extending it until we can estimate the[latex]\\,\\frac{\\text{rise}}{\\text{run}}.[\/latex]<\/p>\n\n<div id=\"Example_04_03_02\" class=\"textbox examples\">\n<div id=\"fs-id2262247\">\n<div id=\"fs-id1575271\">\n<h3>Finding a Line of Best Fit<\/h3>\n<p id=\"fs-id2268675\">Find a linear function that fits the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a> by \u201ceyeballing\u201d a line that seems to fit.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1497780\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1497780\"]\n<p id=\"fs-id1497780\">On a graph, we could try sketching a line. Using the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of<\/p>\n\n<div id=\"fs-id1801316\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}m=\\frac{60}{50}\\hfill \\\\ \\,\\,\\,\\,=1.2\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2651822\">and a <em>y<\/em>-intercept at 30. This gives an equation of<\/p>\n\n<div id=\"fs-id1676754\" class=\"unnumbered aligncenter\">[latex]T\\left(c\\right)=1.2c+30[\/latex]<\/div>\nwhere[latex]\\,c\\,[\/latex]is the number of chirps in 15 seconds, and[latex]\\,T\\left(c\\right)\\,[\/latex]is the temperature in degrees Fahrenheit. The resulting equation is represented in <a class=\"autogenerated-content\" href=\"#Figure_04_03_003\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/c3b09ea16d20f7e7e4ac733ef5fade8c17a5d929\/CNX_Precalc_Figure_02_04_003.jpg\" alt=\"Scatter plot, showing the line of best fit: T(c) = 1.2c + 30. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"432\"> <strong>Figure 3.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1486842\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1720506\">This linear equation can then be used to approximate answers to various questions we might ask about the trend.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1720372\" class=\"bc-section section\">\n<h4>Recognizing Interpolation or Extrapolation<\/h4>\n<p id=\"fs-id1711565\">While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as <strong>interpolation <\/strong>when we predict a value inside the domain and range of the data. The process of <strong>extrapolation <\/strong>is used when we predict a value outside the domain and range of the data.<\/p>\n<p id=\"fs-id2208869\"><a class=\"autogenerated-content\" href=\"#Figure_04_03_004\">(Figure)<\/a> compares the two processes for the cricket-chirp data addressed in <a class=\"autogenerated-content\" href=\"#Example_04_03_02\">(Figure)<\/a>. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.<\/p>\n<p id=\"fs-id2560478\">There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when[latex]\\,x=50,[\/latex]that is in 50 years, the model would not apply because we could not account for factors fifty years in the future.<\/p>\n\n<div id=\"Figure_04_03_004\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/cnx.org\/resources\/fd26d5773978743dded07a498c069ea62e7cc5b1\/CNX_Precalc_Figure_02_04_004.jpg\" alt=\"Scatter plot, showing the line of best fit. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'. The area around the scattered points is enclosed in a box labeled: Interpolation. The area outside of this box is labeled: Extrapolation.\" width=\"487\" height=\"430\"> <strong>Figure 4. <\/strong>Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.[\/caption]\n\n<\/div>\n<div id=\"fs-id2570361\" class=\"textbox key-takeaways\">\n<h3>Interpolation and Extrapolation<\/h3>\n<p id=\"fs-id1633393\">Different methods of making predictions are used to analyze data.<\/p>\n<p id=\"eip-872\">The method of interpolation involves predicting a value inside the domain and\/or range of the data.<\/p>\nThe method of extrapolation involves predicting a value outside the domain and\/or range of the data.\n\nModel breakdown occurs at the point when the model no longer applies.\n\n<\/div>\n<div id=\"Example_04_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1691726\">\n<div id=\"fs-id1620023\">\n<h3>Understanding Interpolation and Extrapolation<\/h3>\n<p id=\"fs-id1842101\">Use the cricket data from <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a> to answer the following questions:<\/p>\n\n<ol id=\"fs-id1713857\" type=\"a\">\n \t<li>Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\n \t<li>Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1570270\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1570270\"]\n<ol id=\"fs-id1570270\" type=\"a\">\n \t<li>The number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data, so would be interpolation. Using our model:\n<div id=\"fs-id1202174\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\text{T (30)}&amp; =&amp; \\text{30 + 1.2(30)}\\hfill \\\\ &amp; =&amp; \\text{66 degrees}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1555289\">Based on the data we have, this value seems reasonable.<\/p>\n<\/li>\n \t<li>The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation because 40 is outside the range of our data. Using our model:\n<div id=\"fs-id1251848\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}40=30+1.2c\\hfill \\\\ 10=1.2c\\hfill \\\\ \\text{ }c\\approx 8.33\\hfill \\end{array}[\/latex]<\/div><\/li>\n<\/ol>\nWe can compare the regions of interpolation and extrapolation using <a class=\"autogenerated-content\" href=\"#Figure_04_03_005\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"485\"]<img src=\"https:\/\/cnx.org\/resources\/3bb7ccc3fc050a07015bdc1ea2861e402262b9fe\/CNX_Precalc_Figure_02_04_005.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps vs. Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'. An additional point is plotted inside of the box to represent an interpolated point. There is another additional point plotted outside of the box to represent an extrapolated point.\" width=\"485\" height=\"429\"> <strong>Figure 5.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1422108\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1768136\">Our model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether below around 50 degrees.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2181223\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_01\">\n<div id=\"fs-id1589415\">\n<p id=\"fs-id1585053\">According to the data from <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a>, what temperature can we predict it is if we counted 20 chirps in 15 seconds?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1685205\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1685205\"]\n<p id=\"fs-id1685205\">[latex]54\u00b0\\text{F}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Finding the Line of Best Fit Using a Graphing Utility<\/h4>\n<p id=\"fs-id2293766\">While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values[footnote]Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.[\/footnote] . One such technique is called least squares regression and can be computed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculators[footnote]For example, <a href=\"http:\/\/www.shodor.org\/unchem\/math\/lls\/leastsq.html\">http:\/\/www.shodor.org\/unchem\/math\/lls\/leastsq.html<\/a>[\/footnote] <sup id=\"footnote-ref3\"><\/sup>. Least squares regression is one means to determine the line that best fits the data, and here we will refer to this method as linear regression.<\/p>\n\n<div id=\"fs-id2627255\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id2251507\"><strong>Given data of input and corresponding outputs from a linear function, find the best fit line using linear regression.<\/strong><\/p>\n\n<ol id=\"fs-id1503997\" type=\"1\">\n \t<li>Enter the input in List 1 (L1).<\/li>\n \t<li>Enter the output in List 2 (L2).<\/li>\n \t<li>On a graphing utility, select Linear Regression (LinReg).<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_03_04\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1632771\">\n<h3>Finding a Least Squares Regression Line<\/h3>\n<p id=\"fs-id2575012\">Find the least squares <span class=\"no-emphasis\">regression line<\/span> using the cricket-chirp data in <a class=\"autogenerated-content\" href=\"#Table_04_03_02\">(Figure)<\/a>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1587366\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1587366\"]\n<ol id=\"fs-id1587366\" type=\"1\">\n \t<li>Enter the input (chirps) in List 1 (L1).<\/li>\n \t<li>Enter the output (temperature) in List 2 (L2). See <a class=\"autogenerated-content\" href=\"#Table_04_03_02\">(Figure)<\/a>.\n<table id=\"Table_04_03_02\" summary=\"This table has two rows and ten columns. The first row is labeled: \u201cL1\u201d and the second is labeled: \u201cL2\u201d. The values in the first row are: 44, 35, 20.4, 33, 31, 35, 18.5, 37, 26. The values in the second row are: 80.5, 70.5, 57, 66, 68, 72, 52, 73.5, 53.\">\n<tbody>\n<tr>\n<td><strong>L1<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>L2<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n \t<li>On a graphing utility, select Linear Regression (LinReg). Using the cricket chirp data from earlier, with technology we obtain the equation:<\/li>\n<\/ol>\n<div id=\"fs-id1618400\" class=\"unnumbered aligncenter\">[latex]T\\left(c\\right)=30.281+1.143c[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1594310\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1695711\">Notice that this line is quite similar to the equation we \u201ceyeballed\u201d but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:<\/p>\n\n<div id=\"fs-id2136214\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}T\\left(30\\right)=30.281+1.143\\left(30\\right)\\hfill \\\\ \\text{ }=64.571\\hfill \\\\ \\text{ }\\approx 64.6\\text{ degrees}\\hfill \\end{array}[\/latex]<\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/518bc40eb0e1068ac46e62e9f2a414854c98e0f6\/CNX_Precalc_Figure_02_04_006.jpg\" alt=\"Scatter plot, showing the line of best fit: T(c) = 30.281 + 1.143c. It is titled 'Cricket Chirps vs. Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"408\"> <strong>Figure 6.<\/strong>[\/caption]\n\nThe graph of the scatter plot with the least squares regression line is shown in <a class=\"autogenerated-content\" href=\"#Figure_04_03_006\">(Figure)<\/a>.\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1592598\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id2181364\"><strong>Will there ever be a case where two different lines will serve as the best fit for the data? <\/strong><\/p>\n<p id=\"fs-id1688806\"><em>No. There is only one best fit line.<\/em><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1459299\" class=\"bc-section section\">\n<h3>Distinguishing Between Linear and Nonlinear Models<\/h3>\n<p id=\"fs-id1586029\">As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <span class=\"no-emphasis\">correlation coefficient<\/span>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a \u201ddiagnostic on\u201d selection to find the correlation coefficient, which mathematicians label as[latex]\\,r\\,[\/latex]The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.<\/p>\n<p id=\"fs-id2674051\">We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of[latex]\\,r\\,[\/latex]and the graph of the data, <a class=\"autogenerated-content\" href=\"#Figure_04_03_007\">(Figure)<\/a> shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.<\/p>\n\n<div id=\"Figure_04_03_007\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/cnx.org\/resources\/f97e72fcb8b9233a23aeebdbb15e21f80163163c\/CNX_Precalc_Figure_02_04_007.jpg\" alt=\"Correlation coefficients values range from -1.0 - 1.0. Collections of dots representing an example of each kind of correlation coefficient are plotted underneath them. The closer to 1.0 the more the points are grouped tightly to form a line in the positive direction. The closer to -1.0 the more the points are grouped tightly to form a line in the negative direction. The closer to 0 the points are very scattered and do not form a line. Several shapes are displayed at the bottom row, none of which are lines, but all of them have values of 0.\" width=\"975\" height=\"434\"> <strong>Figure 7.<\/strong> Plotted data and related correlation coefficients. (credit: \u201cDenisBoigelot,\u201d Wikimedia Commons)[\/caption]\n<h3>Correlation Coefficient<\/h3>\n<div id=\"fs-id1415360\">\n<p id=\"fs-id1589701\">The <strong>correlation coefficient<\/strong> is a value,[latex]\\,r,[\/latex]between \u20131 and 1.<\/p>\n\n<ul id=\"fs-id1963658\">\n \t<li>[latex]r&gt;0\\,[\/latex]suggests a positive (increasing) relationship<\/li>\n \t<li>[latex]r&lt;0\\,[\/latex]suggests a negative (decreasing) relationship<\/li>\n \t<li>The closer the value is to 0, the more scattered the data.<\/li>\n \t<li>The closer the value is to 1 or \u20131, the less scattered the data is.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1367501\">\n<div id=\"fs-id1703090\">\n<h3>Finding a Correlation Coefficient<\/h3>\n<p id=\"fs-id2165511\">Calculate the correlation coefficient for cricket-chirp data in <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a>.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2164982\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2164982\"]\n<p id=\"fs-id2164982\">Because the data appear to follow a linear pattern, we can use technology to calculate[latex]\\,r\\,[\/latex]Enter the inputs and corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation coefficient,[latex]\\,r=0.9509.\\,[\/latex]This value is very close to 1, which suggests a strong increasing linear relationship.<\/p>\n<p id=\"fs-id1154034\">Note: For some calculators, the Diagnostics must be turned \"on\" in order to get the correlation coefficient when linear regression is performed: [2nd]&gt;[0]&gt;[alpha][x\u20131], then scroll to DIAGNOSTICSON.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1840824\" class=\"bc-section section\">\n<h3>Fitting a Regression Line to a Set of Data<\/h3>\n<p id=\"fs-id1685569\">Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.<\/p>\n\n<div id=\"Example_04_03_06\" class=\"textbox examples\">\n<div id=\"fs-id917127\">\n<div id=\"fs-id1827875\">\n<h3>Using a Regression Line to Make Predictions<\/h3>\n<p id=\"fs-id1587319\">Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in <a class=\"autogenerated-content\" href=\"#Table_04_03_03\">(Figure)<\/a>.[footnote]http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html[\/footnote]<sup id=\"footnote-ref4\"><\/sup> Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.<\/p>\n\n<table id=\"Table_04_03_03\" summary=\"This table has two rows and twelve columns. The first row is labeled: \u201cYear\u201d and the second is labeled: \u201cConsumption (billions of gallons)\u201d. The values in the first row are: \u201994, \u201995, \u201996, \u201997, \u201998, \u201999, \u201900, \u201901, \u201902, \u201903, \u201904. The values in the second row are: 113, 116, 118, 119, 123, 125, 126, 128, 131, 133, 136.\">\n<tbody>\n<tr>\n<td style=\"width: 154px\"><strong>Year<\/strong><\/td>\n<td style=\"width: 31px\">'94<\/td>\n<td style=\"width: 31px\">'95<\/td>\n<td style=\"width: 31px\">'96<\/td>\n<td style=\"width: 31px\">'97<\/td>\n<td style=\"width: 31px\">'98<\/td>\n<td style=\"width: 31px\">'99<\/td>\n<td style=\"width: 31px\">'00<\/td>\n<td style=\"width: 31px\">'01<\/td>\n<td style=\"width: 31px\">'02<\/td>\n<td style=\"width: 31px\">'03<\/td>\n<td style=\"width: 31px\">'04<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 154px\"><strong>Consumption (billions of gallons)<\/strong><\/td>\n<td style=\"width: 31px\">113<\/td>\n<td style=\"width: 31px\">116<\/td>\n<td style=\"width: 31px\">118<\/td>\n<td style=\"width: 31px\">119<\/td>\n<td style=\"width: 31px\">123<\/td>\n<td style=\"width: 31px\">125<\/td>\n<td style=\"width: 31px\">126<\/td>\n<td style=\"width: 31px\">128<\/td>\n<td style=\"width: 31px\">131<\/td>\n<td style=\"width: 31px\">133<\/td>\n<td style=\"width: 31px\">136<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThe scatter plot of the data, including the least squares regression line, is shown in <a class=\"autogenerated-content\" href=\"#Figure_04_03_008\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/46361ea50eb0fd022efd04ebd947b65218bcc6f8\/CNX_Precalc_Figure_02_04_008.jpg\" alt=\"Scatter plot, showing the line of best fit. It is titled 'Gas Consumption VS Year'. The x-axis is 'Year After 1994', and the y-axis is 'Gas Consumption (billions of gallons)'. The points are strongly positively correlated and the line of best fit goes through most of the points completely. \" width=\"487\" height=\"384\"> <strong>Figure 8.<\/strong>[\/caption]\n\n&nbsp;\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1588946\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1588946\"]\n<p id=\"fs-id1588946\">We can introduce new input variable,[latex]\\,t,[\/latex]representing years since 1994.<\/p>\n<p id=\"fs-id1565498\">The least squares regression equation is:<\/p>\n\n<div id=\"fs-id2065061\" class=\"unnumbered aligncenter\">[latex]C\\left(t\\right)=113.318+2.209t[\/latex]<\/div>\n<p id=\"fs-id1503420\">Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.<\/p>\n<p id=\"fs-id2786561\">Using this to predict consumption in 2008[latex]\\,\\left(t=14\\right),[\/latex]<\/p>\n\n<div id=\"fs-id1737282\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}C\\left(14\\right)=113.318+2.209\\left(14\\right)\\hfill \\\\ \\text{ }=144.244\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1716786\">The model predicts 144.244 billion gallons of gasoline consumption in 2008.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1799110\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_02\">\n<div id=\"fs-id1549586\">\n<p id=\"fs-id1619902\">Use the model we created using technology in <a class=\"autogenerated-content\" href=\"#Example_04_03_06\">(Figure)<\/a> to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1608557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1608557\"]\n<p id=\"fs-id1608557\">150.871 billion gallons; extrapolation<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1551115\" class=\"precalculus media\">\n<p id=\"fs-id1582076\">Access these online resources for additional instruction and practice with fitting linear models to data.<\/p>\n\n<ul id=\"bulleted\">\n \t<li><a href=\"http:\/\/Openstaxcollege.org\/l\/introregress\">Introduction to Regression Analysis<\/a><\/li>\n \t<li><a href=\"http:\/\/Openstaxcollege.org\/l\/linearregress\">Linear Regression<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1486512\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1862819\">\n \t<li>Scatter plots show the relationship between two sets of data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_01\">(Figure)<\/a>.<\/li>\n \t<li>Scatter plots may represent linear or non-linear models.<\/li>\n \t<li>The line of best fit may be estimated or calculated, using a calculator or statistical software. See <a class=\"autogenerated-content\" href=\"#Example_04_03_02\">(Figure)<\/a>.<\/li>\n \t<li>Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_03\">(Figure)<\/a>.<\/li>\n \t<li>The correlation coefficient,[latex]\\,r,[\/latex]indicates the degree of linear relationship between data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_04\">(Figure)<\/a>.<\/li>\n \t<li>A regression line best fits the data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_05\">(Figure)<\/a>.<\/li>\n \t<li>The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See <a class=\"autogenerated-content\" href=\"#Example_04_03_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id942179\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2598318\">\n<div id=\"fs-id2248686\">\n<p id=\"fs-id1598256\">Describe what it means if there is a model breakdown when using a linear model.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1412205\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1412205\"]\n<p id=\"fs-id1412205\">When our model no longer applies, after some value in the domain, the model itself doesn\u2019t hold.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2638023\">\n<div id=\"fs-id1690731\">\n<p id=\"fs-id2090338\">What is interpolation when using a linear model?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id888239\">\n<div id=\"fs-id940827\">\n<p id=\"fs-id1709982\">What is extrapolation when using a linear model?<\/p>\n\n<\/div>\n<div id=\"fs-id926962\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id926962\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id926962\"]\n<p id=\"fs-id2113896\">We predict a value outside the domain and range of the data.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id918008\">\n<div id=\"fs-id1472127\">\n<p id=\"fs-id894113\">Explain the difference between a positive and a negative correlation coefficient.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2158268\">\n<div>\n<p id=\"fs-id1791770\">Explain how to interpret the absolute value of a correlation coefficient.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1570206\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1570206\"]\n<p id=\"fs-id1570206\">The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1619206\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id2484796\">\n<div id=\"fs-id1540854\">\n<p id=\"fs-id2552133\">A regression was run to determine whether there is a relationship between hours of TV watched per day[latex]\\,\\left(x\\right)\\,[\/latex]and number of sit-ups a person can do[latex]\\,\\left(y\\right).\\,[\/latex]The results of the regression are given below. Use this to predict the number of sit-ups a person who watches 11 hours of TV can do.<\/p>\n\n<div id=\"fs-id1487852\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=ax+b\\hfill \\\\ a=-1.341\\hfill \\\\ b=32.234\\hfill \\\\ \\text{ }r=-0.896\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1541363\">\n<div id=\"fs-id2643682\">\n<p id=\"fs-id1626298\">A regression was run to determine whether there is a relationship between the diameter of a tree ([latex]\\,x,[\/latex]in inches) and the tree\u2019s age ([latex]\\,y,[\/latex]in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches.<\/p>\n\n<div id=\"fs-id1474007\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=ax+b\\hfill \\\\ a=6.301\\hfill \\\\ b=-1.044\\hfill \\\\ \\text{ }r=-0.970\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2159539\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2159539\"]\n<p id=\"fs-id2159539\">61.966 years<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2051997\">For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?<\/p>\n\n<div id=\"fs-id1790165\">\n<div id=\"fs-id2634740\">\n<table id=\"Table_04_03_04\" class=\"unnumbered\" summary=\"This table includes two rows and six columns. The values in the first row are 0, 2, 4, 6, 8, 10. The values in the second row are: -22, -19, -15, -11, -6, -2.\">\n<tbody>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u201322<\/td>\n<td>\u201319<\/td>\n<td>\u201315<\/td>\n<td>\u201311<\/td>\n<td>\u20136<\/td>\n<td>\u20132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2217614\">\n<div id=\"fs-id2097711\">\n<table id=\"Table_04_03_35\" class=\"unnumbered\" summary=\"This table has two rows and six columns. The values in the first row are: 1, 2, 3, 4, 5, 6. The values in the second row are 46, 50, 59, 75, 100, 136.\">\n<tbody>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>46<\/td>\n<td>50<\/td>\n<td>59<\/td>\n<td>75<\/td>\n<td>100<\/td>\n<td>136<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1412608\"]Show Solution[\/reveal-answer][hidden-answer a=\"1412608\"]<img src=\"https:\/\/cnx.org\/resources\/7830e8f085112456858cc05c30f6dddc829d133f\/CNX_Precalc_Figure_02_04_234.jpg\" alt=\"Scatter plot with a collection of points appearing at (1,46); (2,50); (3,59); (4,75); (5, 100); and (6,136); they do not appear linear\">\n<p id=\"fs-id1534066\">No.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1541935\">\n<div id=\"fs-id1374381\">\n<table id=\"Table_04_03_05\" class=\"unnumbered\" summary=\"Two rows and six columns. The values in the first row are: 100, 250, 300, 450, 600, 750. The values in the second row are: 12, 12.6, 13.1, 14, 14.5, 15.2.\">\n<tbody>\n<tr>\n<td>100<\/td>\n<td>250<\/td>\n<td>300<\/td>\n<td>450<\/td>\n<td>600<\/td>\n<td>750<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>12.6<\/td>\n<td>13.1<\/td>\n<td>14<\/td>\n<td>14.5<\/td>\n<td>15.2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id894444\">\n<div id=\"fs-id893652\">\n<table id=\"Table_04_03_06\" class=\"unnumbered\" summary=\"Two rows and six columns. The values in the first row are: 1, 3, 5, 7, 9, 11. The values in the second row are 1, 9, 28, 65, 125, 216.\">\n<tbody>\n<tr>\n<td>1<\/td>\n<td>3<\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>9<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>9<\/td>\n<td>28<\/td>\n<td>65<\/td>\n<td>125<\/td>\n<td>216<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1326815\"]Show Solution[\/reveal-answer][hidden-answer a=\"1326815\"]<img src=\"https:\/\/cnx.org\/resources\/aec19f39026dfb8f3d1266ac042c3bb70e4c730e\/CNX_Precalc_Figure_02_04_236.jpg\" alt=\"Scatterplot with a collection of points at (1,1); (3,9); (5,28); (7,65); (9,125); and (11,216); they do not appear linear\">\n<p id=\"fs-id1542149\">No.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1549099\">\n<div id=\"fs-id2257504\">\n<p id=\"fs-id1536153\">For the following data, draw a scatter plot. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation? Eyeball the line, and estimate the answer.<\/p>\n\n<table id=\"Table_04_03_07\" class=\"unnumbered\" summary=\"This table includes two columns and six rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cPopulation\u201d. The values in the first column are: 1990, 1995, 2000, 2005, 2010. The values in the second column are 11,500; 12,100; 12,700; 13,000; 13,750.\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Population<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>11,500<\/td>\n<\/tr>\n<tr>\n<td>1995<\/td>\n<td>12,100<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>12,700<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>13,000<\/td>\n<\/tr>\n<tr>\n<td>2010<\/td>\n<td>13,750<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2202140\">\n<div id=\"fs-id1737923\">\n<p id=\"fs-id1604883\">For the following data, draw a scatter plot. If we wanted to know when the temperature would reach 28\u00b0F, would the answer involve interpolation or extrapolation? Eyeball the line and estimate the answer.<\/p>\n\n<table id=\"Table_04_03_08\" class=\"unnumbered\" summary=\"This table includes two rows and six columns. The first column is labeled: \u201cTemperature, \u00b0F\u201d and the second is labeled: \u201cTime, seconds\u201d. The values in the first row are 16, 18, 20, 25, 30. The values in the second row are: 46, 50, 54, 55, 62.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>Temperature,\u00b0F<\/strong><\/td>\n<td>16<\/td>\n<td>18<\/td>\n<td>20<\/td>\n<td>25<\/td>\n<td>30<\/td>\n<\/tr>\n<tr>\n<td><strong>Time, seconds<\/strong><\/td>\n<td>46<\/td>\n<td>50<\/td>\n<td>54<\/td>\n<td>55<\/td>\n<td>62<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2078772\"]Show Solution[\/reveal-answer][hidden-answer a=\"2078772\"]<img src=\"https:\/\/cnx.org\/resources\/f57338a3b93fc667a821d10879a31024671db499\/CNX_Precalc_Figure_02_04_238.jpg\" alt=\"Scatterplot with a collection of points at (16,46); (18,50); (20,54); (25,55); and (30,62); they appear nonlinear\">\n<p id=\"fs-id1470010\">Interpolation. About [latex]\\,60\u00b0F.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1475649\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n[caption id=\"\" align=\"aligncenter\" width=\"465\"]<img src=\"https:\/\/cnx.org\/resources\/027e806014d4c7ede73d26b885a3951f322d105e\/CNX_Precalc_Figure_02_04_204ab.jpg\" alt=\"Side-by-side scatter plots. The first is a scattered correlation in the positive direction. The second is a scattered correlation in the negative direction\" width=\"465\" height=\"864\"> <strong>Figure 9.<\/strong>[\/caption]\n\n[caption id=\"\" align=\"aligncenter\" width=\"465\"]<img src=\"https:\/\/cnx.org\/resources\/91207bda2341bb80d0e79a83f80225aa5b14414c\/CNX_Precalc_Figure_02_04_204cd.jpg\" alt=\"Side-by-side scatter plots. The first has a strong negative correlation with all the points spaced out evenly near the top and center, but more spread out near the bottom. The second has a strong positive correlation, with the points more spread out near the bottom and closer together near the center and top.\" width=\"465\" height=\"864\"> <strong>Figure 10.<\/strong>[\/caption]\n<p id=\"fs-id2570326\">For the following exercises, match each scatterplot with one of the four specified correlations in <a class=\"autogenerated-content\" href=\"#Figure_04_03_201\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Figure_04_03_202a\">(Figure)<\/a><strong>.<\/strong><span id=\"fs-id1566729\"><\/span><\/p>\n\n<div id=\"fs-id1284066\">\n<div>\n<p id=\"fs-id1380305\">[latex]r=0.\\text{95}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1619998\">\n<div id=\"fs-id834228\">\n<p id=\"fs-id2477405\">[latex]r=-0.\\text{89}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1395980\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1395980\"]\n<p id=\"fs-id1395980\">[latex]\\text{This value of r indicates a strong negative correlation or slope, so C}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2281428\">\n<div id=\"fs-id1501585\">\n<p id=\"fs-id1738451\">[latex]r=-0.26[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2660448\">\n<div id=\"fs-id1412044\">[latex]r=-0.39[\/latex]<\/div>\n<div id=\"fs-id995908\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id995908\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id995908\"]\n<p id=\"fs-id1708754\">[latex]\\text{This value of r indicates a weak negative correlation, so B}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1691868\">For the following exercises, draw a best-fit line for the plotted data.<\/p>\n\n<div id=\"fs-id1374546\">\n<div id=\"fs-id1694634\"><span id=\"fs-id2094007\"><img src=\"https:\/\/cnx.org\/resources\/6cb41d21268059aa5b04cfcfbb4f1d62fb544070\/CNX_Precalc_02_04_202.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of 4 to 9. The points are at (0,5); (2.1,4.2); (3.5,6); (4.5,6.5); (5.5,6.8); (7,7.4); (8,8.5); (9,8); and (10,9).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id2366654\">\n<div id=\"fs-id2366655\"><img src=\"https:\/\/cnx.org\/resources\/e17241eb73231fa5f6f7a11d8f02329400609555\/CNX_Precalc_Figure_02_04_204.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of -1 to 4. The points are at (0,1.5); (1.5, -0.1); (2.1,1.9); (3.4, 1.5); (4.5,2.5); (5.8,2.2); (6.8,3.8); (7.8,3.6); (8.8,2); and (10,2.4).\"><\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1729040\"]Show Solution[\/reveal-answer][hidden-answer a=\"1729040\"]<img src=\"https:\/\/cnx.org\/resources\/71e838bf549242004a7f1eecfeeb66cde279b324\/CNX_Precalc_Figure_02_04_205.jpg\" alt=\"Scatter plot with domain 0 to 10 and a range from -1 to 4 with the line of best fit drawn going through the points: (0,1.5); (1.5, -0.1); (2.1,1.9); (3.4, 1.5); (4.5,2.5); (5.8,2.2); (6.8,3.8); (7.8,3.6); (8.8,2); and (10,2.4).\">[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1703008\">\n<div id=\"fs-id1703009\"><span id=\"fs-id1630678\"><img src=\"https:\/\/cnx.org\/resources\/f8060b2872a3c3ed4a857f5e8d1113745de760bb\/CNX_Precalc_Figure_02_04_206.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and range of 0 to 7 with the points: (0,7.3); (1,7); (2.2,6); (3.6,7); (4.8,6.2); (5.8,4); (6.6,3.8); (7.9,2.4); (8.8,2); and (10,0.1).\"><img src=\"https:\/\/cnx.org\/resources\/4fe282978ff5bfb9bde48a0492a2e094a3d63d50\/CNX_Precalc_Figure_02_04_208.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of 2 to 6 with the points: (0,2.1); (1,3.9); (2.1,3.6); (3.6,3.9); (4.4,4); (5.6,4.2); (6.8,5); (7.8,5); (9,5.6); and (10,6).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1581440\">\n<div id=\"fs-id1581441\"><span id=\"fs-id1451072\">&nbsp;<\/span><\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1702047\"]Show Solution[\/reveal-answer][hidden-answer a=\"1702047\"]<img src=\"https:\/\/cnx.org\/resources\/cbd36cc651fdbd957dfcd9a64989a9f9053ea0a6\/CNX_Precalc_Figure_02_04_210.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of 2 to 6 and the line of best fit going through the points: (0,2.1); (1,3.9); (2.1,3.6); (3.6,3.9); (4.4,4); (5.6,4.2); (6.8,5); (7.8,5); (9,5.6); and (10,6)\">[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2464699\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<div id=\"fs-id1690887\">\n<div id=\"fs-id1690888\">\n<p id=\"fs-id1695773\">The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given in <a class=\"autogenerated-content\" href=\"#Table_04_03_11\">(Figure)<\/a>.[footnote]Based on data from http:\/\/www.census.gov\/hhes\/socdemo\/education\/data\/cps\/historical\/index.html. Accessed 5\/1\/2014.[\/footnote]<sup id=\"footnote-ref5\"><\/sup> Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the percentage exceed 35%?<\/p>\n\n<table id=\"Table_04_03_11\" summary=\"This table includes two columns and eleven rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cPercent Graduates\u201d. The values in the first column are: 1990, 1992, 1994, 1996, 1998, 2000, 2002, 2004, 2006, 2008. The values in the second column are: 21.3, 21.4, 22.2, 23.6, 24.4, 25.6, 26.7, 27.7, 28, 29.4\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Percent Graduates<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>21.3<\/td>\n<\/tr>\n<tr>\n<td>1992<\/td>\n<td>21.4<\/td>\n<\/tr>\n<tr>\n<td>1994<\/td>\n<td>22.2<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>23.6<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>24.4<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>25.6<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>26.7<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>27.7<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>28<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>29.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1798367\">\n<div id=\"fs-id1565618\">\n<p id=\"fs-id1565619\">The U.S. import of wine (in hectoliters) for several years is given in <a class=\"autogenerated-content\" href=\"#Table_04_03_12\">(Figure)<\/a>. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will imports exceed 12,000 hectoliters?<\/p>\n\n<table id=\"Table_04_03_12\" summary=\"This table includes two columns and eleven rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cImports\u201d. The values in the first column are: 1992, 1994, 1996, 1998, 2000, 2002, 2004, 2006, 2008, 2009. The values in the second solumn are: 2665, 2688, 3565, 4129, 4584, 5655, 6549, 7950, 8487, 9462\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Imports<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1992<\/td>\n<td>2665<\/td>\n<\/tr>\n<tr>\n<td>1994<\/td>\n<td>2688<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>3565<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>4129<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>4584<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>5655<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>6549<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>7950<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>8487<\/td>\n<\/tr>\n<tr>\n<td>2009<\/td>\n<td>9462<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1629450\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1629450\"]\n<p id=\"fs-id1629450\">Yes, trend appears linear because[latex]\\,r=0.\\text{985}\\,[\/latex]and will exceed 12,000 near midyear, 2016, 24.6 years since 1992.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1487111\">\n<div id=\"fs-id1900537\">\n<p id=\"fs-id1900538\"><a class=\"autogenerated-content\" href=\"#Table_04_03_13\">(Figure)<\/a> shows the year and the number of people unemployed in a particular city for several years. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the number of unemployed reach 5?<\/p>\n\n<table id=\"Table_04_03_13\" summary=\"This table includes two columns and eleven rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cNumber Unemployed\u201d. The values in the first column are: 1990, 1992, 1994, 1996, 1998, 2000, 2002, 2004, 2006, 2008. The values in the second column are: 750, 670, 650, 605, 550, 510, 460, 420, 380, 320.\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Number Unemployed<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>750<\/td>\n<\/tr>\n<tr>\n<td>1992<\/td>\n<td>670<\/td>\n<\/tr>\n<tr>\n<td>1994<\/td>\n<td>650<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>605<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>550<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>510<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>460<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>420<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>380<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>320<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1390394\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id2368077\">For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.<\/p>\n\n<div id=\"fs-id1712789\">\n<div id=\"fs-id2041879\">\n<table id=\"Table_04_03_14\" class=\"unnumbered\" summary=\"This table includes two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 8, 15, 26, 31, 56. The values in the second row are: 23, 41, 53, 72, 103.\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>15<\/td>\n<td>26<\/td>\n<td>31<\/td>\n<td>56<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>23<\/td>\n<td>41<\/td>\n<td>53<\/td>\n<td>72<\/td>\n<td>103<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1689902\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1689902\"]\n<p id=\"fs-id1689902\">[latex]y=\\text{1}.\\text{64}0x+\\text{13}.\\text{8}00,[\/latex][latex]r=0.\\text{987}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1595978\">\n<div id=\"fs-id1595979\">\n<table id=\"Table_04_03_15\" class=\"unnumbered\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 5, 7, 10, 12, 15. The values in the second row are: 4, 12, 17, 22, 24\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>10<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>4<\/td>\n<td>12<\/td>\n<td>17<\/td>\n<td>22<\/td>\n<td>24<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1474083\">\n<div id=\"fs-id1429608\">\n<table id=\"Table_04_03_16\" class=\"unnumbered\" summary=\"This table has two columns and sixteen rows. The first column is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first column are: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18. The values in the second column are: 21.9, 22.22, 22.74, 22.26, 20.78, 17.6, 16.52, 18.54, 15.76, 13.68, 14.1, 14.02, 11.94, 12.76, 11.28, 9.1.\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>3<\/td>\n<td>21.9<\/td>\n<td>10<\/td>\n<td>18.54<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>22.22<\/td>\n<td>11<\/td>\n<td>15.76<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>22.74<\/td>\n<td>12<\/td>\n<td>13.68<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>22.26<\/td>\n<td>13<\/td>\n<td>14.1<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>20.78<\/td>\n<td>14<\/td>\n<td>14.02<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>17.6<\/td>\n<td>15<\/td>\n<td>11.94<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>16.52<\/td>\n<td>16<\/td>\n<td>12.76<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1843470\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1843470\"]\n<p id=\"fs-id1843470\">[latex]y=-0.962x+26.86, r=-0.965[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1862833\">\n<div id=\"fs-id1862834\">\n<table id=\"Table_04_03_17\" class=\"unnumbered\" summary=\"This table has two columns and eleven rows. The first column is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first column are: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. The values in the second column are: 44.8, 43.1, 38.8, 39, 38, 32.7, 30.1, 29.3, 27, 25.8\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>4<\/td>\n<td>44.8<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>43.1<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>38.8<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>39<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>38<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>32.7<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>30.1<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>29.3<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>27<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>25.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1621529\">\n<div id=\"fs-id1713613\">\n<table id=\"Table_04_03_18\" class=\"unnumbered\" summary=\"This table has two rows and seven columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 21, 25, 30, 31, 40, 50. The values in the second row are: 17, 11, 2, -1, -18, -40.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>21<\/td>\n<td>25<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>40<\/td>\n<td>50<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>17<\/td>\n<td>11<\/td>\n<td>2<\/td>\n<td>\u20131<\/td>\n<td>\u201318<\/td>\n<td>\u201340<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1704740\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1704740\"]\n<p id=\"fs-id1704740\">[latex]y=-\\text{1}.\\text{981}x+\\text{6}0.\\text{197;}[\/latex][latex]r=-0.\\text{998}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1301552\">\n<div id=\"fs-id2061731\">\n<table id=\"Table_04_03_19\" class=\"unnumbered\" summary=\"This table has two columns and seven rows. The first column is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first column are: 100, 80, 60, 55, 40, 20. The values in the second column are: 2000, 1798, 1589, 1580, 1390, 1202.\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>100<\/td>\n<td>2000<\/td>\n<\/tr>\n<tr>\n<td>80<\/td>\n<td>1798<\/td>\n<\/tr>\n<tr>\n<td>60<\/td>\n<td>1589<\/td>\n<\/tr>\n<tr>\n<td>55<\/td>\n<td>1580<\/td>\n<\/tr>\n<tr>\n<td>40<\/td>\n<td>1390<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>1202<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1549953\">\n<div id=\"fs-id1450966\">\n<table id=\"Table_04_03_20\" class=\"unnumbered\" summary=\"This table has two rows and seven columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 900, 988, 1000, 1010, 1200, 1205. The values in the second row are: 70, 80, 82, 84, 105, 108.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>900<\/td>\n<td>988<\/td>\n<td>1000<\/td>\n<td>1010<\/td>\n<td>1200<\/td>\n<td>1205<\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex]<\/td>\n<td>70<\/td>\n<td>80<\/td>\n<td>82<\/td>\n<td>84<\/td>\n<td>105<\/td>\n<td>108<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1475836\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1475836\"]\n<p id=\"fs-id1475836\">[latex]y=0.\\text{121}x-38.841,r=0.998[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2040607\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id2066641\">\n<div id=\"fs-id1701380\">\n<p id=\"fs-id1701381\">Graph[latex]\\,f\\left(x\\right)=0.5x+10.\\,[\/latex]Pick a set of five ordered pairs using inputs[latex]\\,x=-2,\\text{1},\\text{5},\\text{6},\\text{9}\\,[\/latex]and use linear regression to verify that the function is a good fit for the data.<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1334128\">\n<p id=\"fs-id1678235\">Graph[latex]\\,f\\left(x\\right)=-2x-10.\\,[\/latex]Pick a set of five ordered pairs using inputs[latex]\\,x=-2,\\text{1},\\text{5},\\text{6},\\text{9}\\,[\/latex]and use linear regression to verify the function.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1452340\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1452340\"]\n<p id=\"fs-id1452340\">[latex]\\left(-2,-6\\right),\\left(1,\\text{\u221212}\\right),\\left(5,-20\\right),\\left(6,\\text{\u221222}\\right),\\left(9,\\text{\u221228}\\right);\\,[\/latex]Yes, the function is a good fit.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2476616\">For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years:<\/p>\n\n<div id=\"fs-id2084651\" class=\"unnumbered aligncenter\">[latex]\\left(\\text{46},600\\right),\\left(\\text{48},\\text{55}0\\right),\\left(50,505\\right),\\left(\\text{52},\\text{54}0\\right),\\left(\\text{54},\\text{495}\\right).[\/latex]<\/div>\n<div id=\"fs-id1288856\">\n<div id=\"fs-id1288858\">\n<p id=\"fs-id2097965\">Use linear regression to determine a function[latex]\\,P\\,[\/latex]where the profit in thousands of dollars depends on the number of units sold in hundreds.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2178129\">\n<div id=\"fs-id2178130\">\n<p id=\"fs-id2406756\">Find to the nearest tenth and interpret the <em>x<\/em>-intercept.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2490923\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2490923\"]\n<p id=\"fs-id2490923\">[latex]\\left(\\text{189}.8,0\\right)\\,[\/latex]If 18,980 units are sold, the company will have a profit of zero dollars.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1841800\">\n<div id=\"fs-id1579613\">\n<p id=\"fs-id1579614\">Find to the nearest tenth and interpret the <em>y<\/em>-intercept.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2429851\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id2673753\">For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years:<\/p>\n<p id=\"fs-id2459368\">[latex]\\left(\\text{25}00,2000\\right),\\left(\\text{265}0,2001\\right),\\left(3000,2003\\right),\\left(\\text{35}00,2006\\right),\\left(\\text{42}00,2010\\right)[\/latex]<\/p>\n\n<div id=\"fs-id2651915\">\n<div id=\"fs-id2111924\">\n<p id=\"fs-id2111925\">Use linear regression to determine a function[latex]\\,y,[\/latex]where the year depends on the population. Round to three decimal places of accuracy.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1794393\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1794393\"]\n<p id=\"fs-id1794393\">[latex]y=0.00587x+\\text{1985}.4\\text{1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2621050\">\n<div id=\"fs-id2621051\">\n<p id=\"fs-id1469913\">Predict when the population will hit 8,000.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1901427\">For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:<\/p>\n<p id=\"fs-id1798995\">[latex]\\left(\\text{46},\\text{25}0\\right),\\left(\\text{48},\\text{3}05\\right),\\left(50,\\text{35}0\\right),\\left(\\text{52},\\text{39}0\\right),\\left(\\text{54},\\text{41}0\\right).[\/latex]<\/p>\n\n<div id=\"fs-id2202291\">\n<div id=\"fs-id2634058\">\n<p id=\"fs-id2634059\">Use linear regression to determine a function <em>y<\/em>, where the profit in thousands of dollars depends on the number of units sold in hundreds.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2523562\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2523562\"]\n<p id=\"fs-id2523562\">[latex]y=\\text{2}0.\\text{25}x-\\text{671}.\\text{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2095664\">\n<div id=\"fs-id2095665\">\n<p id=\"fs-id2522917\">Predict when the profit will exceed one million dollars.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2227780\">For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years:<\/p>\n<p id=\"fs-id2227782\">[latex]\\left(\\text{46},\\text{25}0\\right),\\left(\\text{48},\\text{225}\\right),\\left(50,\\text{2}05\\right),\\left(\\text{52},\\text{18}0\\right),\\left(\\text{54},\\text{165}\\right).[\/latex]<\/p>\n\n<div id=\"fs-id2575116\">\n<div id=\"fs-id1592705\">\n<p id=\"fs-id1592706\">Use linear regression to determine a function <em>y<\/em>, where the profit in thousands of dollars depends on the number of units sold in hundreds.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2591902\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2591902\"]\n<p id=\"fs-id2591902\">[latex]y=-\\text{1}0.\\text{75}x+\\text{742}.\\text{5}0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2395926\">\n<div id=\"fs-id2395927\">\n<p id=\"fs-id2395928\">Predict when the profit will dip below the $25,000 threshold.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3638138\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id2521178\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f48a9644-4329-4387-9b38-4ac039f12570\">Linear Functions<\/a><\/h4>\n<div id=\"fs-id2806383\">\n<div id=\"fs-id2806384\">\n<p id=\"fs-id2477392\">Determine whether the algebraic equation is linear.[latex]\\,2x+3y=7[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2095696\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2095696\"]\n<p id=\"fs-id2095696\">Yes<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2779858\">\n<div id=\"fs-id2779859\">\n<p id=\"fs-id2779860\">Determine whether the algebraic equation is linear.[latex]\\,6{x}^{2}-y=5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2372659\">\n<div id=\"fs-id2372660\">\n<p id=\"fs-id2372662\">Determine whether the function is increasing or decreasing.<\/p>\n<p id=\"fs-id2060828\">[latex]f\\left(x\\right)=7x-2[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1625952\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1625952\"]\n<p id=\"fs-id1625952\">Increasing<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1625955\">\n<div id=\"fs-id2755050\">\n<p id=\"fs-id2755051\">Determine whether the function is increasing or decreasing.<\/p>\n<p id=\"fs-id2490966\">[latex]g\\left(x\\right)=-x+2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2275942\">\n<div id=\"fs-id2275943\">\n<p id=\"fs-id2275944\">Given each set of information, find a linear equation that satisfies the given conditions, if possible.<\/p>\n<p id=\"fs-id2560758\">Passes through[latex]\\,\\left(\\text{7},\\text{5}\\right)\\,[\/latex]and[latex]\\,\\left(\\text{3},\\text{17}\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2589261\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2589261\"]\n<p id=\"fs-id2589261\">[latex]y=-\\text{3}x+\\text{26}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2523185\">\n<div id=\"fs-id2241203\">\n<p id=\"fs-id2241204\">Given each set of information, find a linear equation that satisfies the given conditions, if possible.<\/p>\n<p id=\"fs-id1726321\"><em>x<\/em>-intercept at[latex]\\,\\left(\\text{6},0\\right)\\,[\/latex]and <em>y<\/em>-intercept at[latex]\\,\\left(0,\\text{1}0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2748158\">\n<div id=\"fs-id2748160\">\n<p id=\"fs-id2748161\">Find the slope of the line shown in the graph.<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/603f3fb474ccfdcae6d7e540b637877ec40c0507\/CNX_Precalc_Figure_02_04_211.jpg\" alt=\"This is a graph of an increasing line with a y-intercept of -3 and x-intercept of 1 on an x, y coordinate plane. The x and y-axis range from -6 to 6.\">\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2555716\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2555716\"]\n<p id=\"fs-id2555716\">3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2555719\">\n<div id=\"fs-id2571097\">\n<p id=\"fs-id2571098\">Find the slope of the line graphed.<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/3b7ee2fb2ebdee991c1f842fdb4e7afcbbeead9f\/CNX_Precalc_Figure_02_04_212.jpg\" alt=\"This is a graph of a line with a y-intercept of -2 and no x-intercepts on an x, y coordinate plane. The x- and y-axis range from -6 to 6\">\n\n<\/div>\n<\/div>\n<div id=\"fs-id2799886\">\n<div id=\"fs-id2799888\">\n<p id=\"fs-id1433211\">Write an equation in slope-intercept form for the line shown.<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/5863d8fbf42fecda8b272b7bc10a9e3533ddd82d\/CNX_Precalc_Figure_02_04_213.jpg\" alt=\"This is a graph of a line with a y-intercept of -2 and x-intercept of 1 on an x, y coordinate plane. The x- and y-axis both range from -6 to 6.\">\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2134830\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2134830\"]\n<p id=\"fs-id2134830\">[latex]y=\\text{2}x-\\text{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2568491\">\n<div id=\"fs-id2568492\">\n<p id=\"fs-id1828687\">Does the following table represent a linear function? If so, find the linear equation that models the data.<\/p>\n\n<table id=\"Table_04_03_21\" class=\"unnumbered\" summary=\"This table has two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cg(x)\u201d. The values in the first row are: -4, 0, 2, 10. The values in the second row are: 18, -2, -12, -52.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>\u20134<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td><strong><em>g(x)<\/em><\/strong><\/td>\n<td>18<\/td>\n<td>\u20132<\/td>\n<td>\u201312<\/td>\n<td>\u201352<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2193041\">\n<div id=\"fs-id2193042\">\n<p id=\"fs-id2193043\">Does the following table represent a linear function? If so, find the linear equation that models the data.<\/p>\n\n<table id=\"Table_04_03_22\" class=\"unnumbered\" summary=\"This table has two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201c\u201d and the second is labeled: \u201cg(x)\u201d. The values in the first row are: 6, 8, 12, 26. The values in the second row are: -8, -12, -18, -46.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>12<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong><em>g(x)<\/em><\/strong><\/td>\n<td>\u20138<\/td>\n<td>\u201312<\/td>\n<td>\u201318<\/td>\n<td>\u201346<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2113224\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2113224\"]\n<p id=\"fs-id2113224\">Not linear.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2241122\">\n<div id=\"fs-id2241123\">\n<p id=\"fs-id2241124\">On June 1<sup>st<\/sup>, a company has $4,000,000 profit. If the company then loses 150,000 dollars per day thereafter in the month of June, what is the company\u2019s profit <em>n<sup>th<\/sup><\/em>day after June 1<sup>st<\/sup>?<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2078767\">For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:<\/p>\n\n<div id=\"fs-id2468288\">\n<div id=\"fs-id2468289\">\n<p id=\"fs-id2468290\">[latex]\\begin{array}{c}2x-6y=12\\\\ -x+3y=1\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2757779\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2757779\"]\n<p id=\"fs-id2757779\">parallel<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2757782\">\n<div id=\"fs-id2637956\">\n<p id=\"fs-id2637957\">[latex]\\begin{array}{c}y=\\frac{1}{3}x-2\\\\ 3x+y=-9\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2616282\">For the following exercises, find the <em>x<\/em>- and <em>y<\/em>- intercepts of the given equation<\/p>\n\n<div id=\"fs-id2489259\">\n<div id=\"fs-id2489260\">\n<p id=\"fs-id2489262\">[latex]7x+9y=-63[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2459233\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2459233\"]\n<p id=\"fs-id2459233\">[latex]\\left(\u20139,0\\right);\\left(0,\u20137\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2779958\">\n<div id=\"fs-id2779959\">\n<p id=\"fs-id1600522\">[latex]f\\left(x\\right)=2x-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2128502\">For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?<\/p>\n\n<div id=\"fs-id1828556\">\n<div id=\"fs-id1828557\">\n<p id=\"fs-id1828558\">Line 1: Passes through[latex]\\,\\left(5,11\\right)\\,[\/latex]and[latex]\\,\\left(10,1\\right)[\/latex]<\/p>\n<p id=\"fs-id1623644\">Line 2: Passes through[latex]\\,\\left(-1,3\\right)\\,[\/latex]and[latex]\\,\\left(-5,11\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1718504\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1718504\"]\n<p id=\"fs-id1718504\">Line 1:[latex]\\,m=-2;[\/latex]Line 2:[latex]\\,m=-2;[\/latex]Parallel<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id889716\">\n<div id=\"fs-id889717\">\n<p id=\"fs-id889718\">Line 1: Passes through[latex]\\,\\left(8,-10\\right)\\,[\/latex]and[latex]\\,\\left(0,-26\\right)[\/latex]<\/p>\n<p id=\"fs-id1598366\">Line 2: Passes through[latex]\\,\\left(2,5\\right)\\,[\/latex]and[latex]\\,\\left(4,4\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2420694\">\n<div id=\"fs-id2480225\">\n<p id=\"fs-id2480226\">Write an equation for a line perpendicular to[latex]\\,f\\left(x\\right)=5x-1\\,[\/latex]and passing through the point (5, 20).<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2202404\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2202404\"]\n<p id=\"fs-id2202404\">[latex]y=-0.2x+21[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1675129\">\n<div id=\"fs-id2779876\">\n<p id=\"fs-id2779877\">Find the equation of a line with a <em>y<\/em>- intercept of[latex]\\,\\left(0,2\\right)\\,[\/latex]and slope[latex]\\,-\\frac{1}{2}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1694239\">\n<div id=\"fs-id1694240\">\n<p id=\"fs-id1694242\">Sketch a graph of the linear function[latex]\\,f\\left(t\\right)=2t-5.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1798839\"]Show Solution[\/reveal-answer][hidden-answer a=\"1798839\"]<img src=\"https:\/\/cnx.org\/resources\/09f3d933b61b5a18631e559c4be70c86977de017\/CNX_Precalc_Figure_02_04_214.jpg\" alt=\"This is a graph of f of t = 2 times t minus 5 on a x, y coordinate plane. The x-axis ranges from -4 to 6 and the y-axis ranges from -6 to 6. The curve is an increasing linear function that goes through the points (0,-5) and (2.5,0). \">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2489570\">\n<div id=\"fs-id2489571\">\n<p id=\"fs-id1966917\">Find the point of intersection for the 2 linear functions:[latex]\\,\\begin{array}{c}x=y+6\\\\ 2x-y=13\\end{array}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2062262\">\n<div id=\"fs-id2062263\">\n<p id=\"fs-id2395879\">A car rental company offers two plans for renting a car.<\/p>\n<p id=\"fs-id2395882\">Plan A: 25 dollars per day and 10 cents per mile<\/p>\n<p id=\"fs-id2486072\">Plan B: 50 dollars per day with free unlimited mileage<\/p>\nHow many miles would you need to drive for plan B to save you money?\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1093901\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1093901\"]\n<p id=\"fs-id1093901\">More than 250<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1093904\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/dcf784dc-acd1-46a8-b5b0-cd5d8673b83a\">Modeling with Linear Functions<\/a><\/h4>\n<div id=\"fs-id1293909\">\n<div id=\"fs-id1293910\">\n<p id=\"fs-id1293911\">Find the area of a triangle bounded by the <em>y<\/em> axis, the line[latex]\\,f\\left(x\\right)=10-2x,[\/latex]and the line perpendicular to[latex]\\,f\\,[\/latex]that passes through the origin.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2063497\">\n<div id=\"fs-id2063498\">\n<p id=\"fs-id2063499\">A town\u2019s population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2276060\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2276060\"]\n<p id=\"fs-id2276060\">118,000<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1843378\">\n<div id=\"fs-id1843379\">\n<p id=\"fs-id1843380\">The number of people afflicted with the common cold in the winter months dropped steadily by 50 each year since 2004 until 2010. In 2004, 875 people were inflicted.<\/p>\n<p id=\"fs-id1484577\">Find the linear function that models the number of people afflicted with the common cold <em>C<\/em> as a function of the year,[latex]\\,t.\\,[\/latex]When will no one be afflicted?<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2748176\">For the following exercises, use the graph in <a class=\"autogenerated-content\" href=\"#Figure_04_03_202\">(Figure)<\/a> showing the profit,[latex]\\,y,[\/latex]in thousands of dollars, of a company in a given year,[latex]\\,x,[\/latex]where[latex]\\,x\\,[\/latex]represents years since 1980.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"291\"]<img src=\"https:\/\/cnx.org\/resources\/67e128c421b4c05cd0a4b2e5c6acdf9b3b796f28\/CNX_Precalc_Figure_02_04_215.jpg\" alt=\"This graph shows profits starting at 1985 at $10,000 and ending at 2005 at $4,000. The x-axis ranges from 0 to 30 in intervals of 5 and the y \u2013axis goes from 0 to 12,000 in intervals of 2,000.\" width=\"291\" height=\"254\"> <strong>Figure 11.<\/strong>[\/caption]\n\n<div id=\"fs-id1429436\">\n<div id=\"fs-id1429437\">\n<p id=\"fs-id2662623\">Find the linear function <em>y<\/em>, where <em>y<\/em> depends on[latex]\\,x,[\/latex]the number of years since 1980.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2637049\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2637049\"]\n<p id=\"fs-id2637049\">[latex]y=-\\text{3}00x+\\text{11},\\text{5}00[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2527720\">\n<div id=\"fs-id2527721\">\n<p id=\"fs-id2527722\">Find and interpret the <em>y<\/em>-intercept.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2552031\">For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.<\/p>\n\n<div id=\"fs-id2518478\">\n<div id=\"fs-id2518479\">\n<p id=\"fs-id2518480\">Assume the population is changing linearly.<\/p>\n\n<ol id=\"fs-id2518483\" type=\"a\">\n \t<li>How much did the population grow between the year 2004 and 2012?<\/li>\n \t<li>What is the average population growth per year?<\/li>\n \t<li>Find an equation for the population, <em>P<\/em>, of the school <em>t<\/em> years after 2004.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2468246\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2468246\"]\n<p id=\"fs-id2468246\">a) 800 b) 100 students per year c)[latex]\\,P\\left(t\\right)=\\text{1}00t+\\text{17}00[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id888455\">For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.<\/p>\n\n<div id=\"fs-id2424214\">\n<div id=\"fs-id2424215\">\n<p id=\"fs-id888457\">Find a formula for the moose population,[latex]\\,P.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2411048\">\n<div id=\"fs-id1016508\">\n<p id=\"fs-id1016509\">What does your model predict the moose population to be in 2020?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1016513\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1016513\"]\n<p id=\"fs-id1016513\">18,500<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2637890\">For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in <a class=\"autogenerated-content\" href=\"#Table_04_03_23\">(Figure)<\/a>. Assume that the house values are changing linearly.<\/p>\n\n<table id=\"Table_04_03_23\" summary=\"This table has three rows and three columns. The first column is labeled: \u201cYear\u201d the second column is labeled: \u201cPima Central\u201d and the third column is labeled: \u201cEast Valley\u201d. The values for the first column are: 1970 and 2010. The values for the second column are: 32,000 and 85,000. The values for the third column are: 120,250 and 150,000.\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Pima Central<\/th>\n<th>East Valley<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1970<\/td>\n<td>32,000<\/td>\n<td>120,250<\/td>\n<\/tr>\n<tr>\n<td>2010<\/td>\n<td>85,000<\/td>\n<td>150,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1802777\">\n<div id=\"fs-id1802778\">\n<p id=\"fs-id1802779\">In which subdivision have home values increased at a higher rate?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1855668\">\n<div id=\"fs-id1855669\">\n<p id=\"fs-id1855670\">If these trends were to continue, what would be the median home value in Pima Central in 2015?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1794327\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1794327\"]\n<p id=\"fs-id1794327\">$91,625<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2267333\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/e9d5f844-6760-4f85-8f79-67292a6703dc\">Fitting Linear Models to Data<\/a><\/h4>\n<div id=\"fs-id2395889\">\n<div id=\"fs-id2395890\">\n<p id=\"fs-id2395891\">Draw a scatter plot for the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_24\">(Figure)<\/a>. Then determine whether the data appears to be linearly related.<\/p>\n\n<table id=\"Table_04_03_24\" summary=\"This table shows two rows and six columns. The values in the first row are: 0, 2, 4, 6, 8, 10. The values in the second row are: -105, -50, 1, 55, 105, 160.\">\n<tbody>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u2013105<\/td>\n<td>\u201350<\/td>\n<td>1<\/td>\n<td>55<\/td>\n<td>105<\/td>\n<td>160<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1739382\">\n<div id=\"fs-id1739383\">\n<p id=\"fs-id1739384\">Draw a scatter plot for the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_25\">(Figure)<\/a>. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation?<\/p>\n\n<table id=\"Table_04_03_25\" summary=\"This table has two columns and six rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cPopulation\u201d. The values in the first column are: 1990, 1995, 2000, 2005, 2010. The values in the second column are: 5,600; 5,950; 6,300; 6,600; 6,900.\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Population<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>5,600<\/td>\n<\/tr>\n<tr>\n<td>1995<\/td>\n<td>5,950<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>6,300<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>6,600<\/td>\n<\/tr>\n<tr>\n<td>2010<\/td>\n<td>6,900<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2208906\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2208906\"]\n<p id=\"fs-id2208906\">Extrapolation<img src=\"https:\/\/cnx.org\/resources\/8beebd5bb1c6ca6c8de43339354cf3293128697c\/CNX_Precalc_Figure_02_04_226.jpg\" alt=\"Scatter plot with the points (1990,5600); (1995,5950); (2000,6300); (2005,6600); and (2010,6900).\"><\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2267039\">\n<div id=\"fs-id2267040\">\n<p id=\"fs-id2267041\">Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in <a class=\"autogenerated-content\" href=\"#Table_04_03_26\">(Figure)<\/a>. Plot the points, then sketch a line that fits the data.<\/p>\n\n<table id=\"Table_04_03_26\" summary=\"This table shows two columns and nine rows. The first column is labeled: \u201cPredicted\u201d and the second is labeled: \u201cActual\u201d. The values in the first column are: 6, 7, 7, 8, 7, 9, 10, 10. The values in the second column are: 6, 7, 8, 8, 9, 10, 10, 9.\">\n<thead>\n<tr>\n<th><strong>Predicted<\/strong><\/th>\n<th><strong>Actual<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>6<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2578689\">\n<div id=\"fs-id2578690\">\n<p id=\"fs-id2578691\">Draw a best-fit line for the plotted data.<\/p>\n<img src=\"https:\/\/cnx.org\/resources\/efcc88d43851ec9b3dd4b1fa62e1491fc2a314ad\/CNX_Precalc_Figure_02_04_216.jpg\" alt=\"Scatter plot of the points: (2,78); (4,81); (6,85); (8,90); and (10,99).\">\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2160490\"]Show Solution[\/reveal-answer][hidden-answer a=\"2160490\"]<img src=\"https:\/\/cnx.org\/resources\/7b9b0a87d04e3bb08bc86beddd2023daa447fb01\/CNX_Precalc_Figure_02_04_217.jpg\" alt=\"Scatter plot of: (2,78); (4,81); (6,85); (8,90); and (10,99) and the line of best fit running through these points. The line of best fit goes through most of the points.\">\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2755468\">For the following exercises, consider the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_27\">(Figure)<\/a>, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year.<\/p>\n\n<table id=\"Table_04_03_27\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cYear\u201d and the second is labeled: \u201cPercent Graduates\u201d. The values in the first row are: 2000, 2002, 2005, 2007, 2010. The values in the second row are: 6.5, 7.0, 7.4, 8.2, 9.0\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2000<\/td>\n<td>2002<\/td>\n<td>2005<\/td>\n<td>2007<\/td>\n<td>2010<\/td>\n<\/tr>\n<tr>\n<td><strong>Percent Graduates<\/strong><\/td>\n<td>6.5<\/td>\n<td>7.0<\/td>\n<td>7.4<\/td>\n<td>8.2<\/td>\n<td>9.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id2489600\">\n<div id=\"fs-id2489245\">\n<p id=\"fs-id2489246\">Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2552236\">\n<div id=\"fs-id2552237\">\n<p id=\"fs-id2552238\">In what year will the percentage exceed 12%?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2448110\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2448110\"]\n<p id=\"fs-id2448110\">2023<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1351720\">Based on the set of data given in <a class=\"autogenerated-content\" href=\"#Table_04_03_28\">(Figure)<\/a>, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.<\/p>\n\n<table id=\"Table_04_03_28\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values of the first row are: 17, 20, 23, 26, 29. The values of the second row are: 15, 25, 31, 37, 40.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>17<\/td>\n<td>20<\/td>\n<td>23<\/td>\n<td>26<\/td>\n<td>29<\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex]<\/td>\n<td>15<\/td>\n<td>25<\/td>\n<td>31<\/td>\n<td>37<\/td>\n<td>40<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2674532\">\n<div id=\"fs-id2164258\">\n<p id=\"fs-id2164259\">Based on the set of data given in <a class=\"autogenerated-content\" href=\"#Table_04_03_29\">(Figure)<\/a>, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.<\/p>\n\n<table id=\"Table_04_03_29\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values of the first row are: 10, 12, 15, 18, 20. The values of the second row are: 36, 34, 30, 28, 22.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>10<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<td>18<\/td>\n<td>20<\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex]<\/td>\n<td>36<\/td>\n<td>34<\/td>\n<td>30<\/td>\n<td>28<\/td>\n<td>22<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2489280\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2489280\"]\n<p id=\"fs-id2489280\">[latex]y=-1.294x+49.412; r=-0.974[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2589276\">For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years:<\/p>\n<p id=\"fs-id1802875\">[latex]\\left(\\text{3,6}00,2000\\right);\\left(\\text{4,}000,2001\\right);\\left(\\text{4,7}00,2003\\right);\\left(\\text{6,}000,2006\\right)[\/latex]<\/p>\n\n<div id=\"fs-id1967151\">\n<div id=\"fs-id1967152\">\n<p id=\"fs-id1967153\">Use linear regression to determine a function[latex]\\,y,[\/latex]where the year depends on the population, to three decimal places of accuracy.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2574347\">\n<div id=\"fs-id2574348\">\n<p id=\"fs-id2574350\">Predict when the population will hit 12,000.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2275847\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2275847\"]\n<p id=\"fs-id2275847\">2027<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2275850\">\n<div id=\"fs-id2275852\">\n<p id=\"fs-id2275853\">What is the correlation coefficient for this model to three decimal places of accuracy?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1505691\">\n<div id=\"fs-id1505692\">\n<p id=\"fs-id1505693\">According to the model, what is the population in 2014?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1505698\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1505698\"]\n<p id=\"fs-id1505698\">7,660<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1505702\" class=\"practice-test\">\n<h3>Chapter Practice Test<\/h3>\n<div id=\"fs-id2560601\">\n<div id=\"fs-id2560602\">\n<p id=\"fs-id2560603\">Determine whether the following algebraic equation can be written as a linear function.[latex]\\,2x+3y=7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1803014\">\n<div id=\"fs-id1803015\">\n<p id=\"fs-id1803016\">Determine whether the following function is increasing or decreasing.[latex]\\,f\\left(x\\right)=-2x+5[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2176627\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2176627\"]\n<p id=\"fs-id2176627\">Decreasing<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2560692\">\n<div id=\"fs-id2560693\">\n<p id=\"fs-id2560694\">Determine whether the following function is increasing or decreasing.[latex]f\\left(x\\right)=7x+9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2424337\">\n<div id=\"fs-id2424338\">\n<p id=\"fs-id2489227\">Find a linear equation that passes through (5, 1) and (3, \u20139), if possible.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2489233\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2489233\"]\n<p id=\"fs-id2489233\">[latex]y=5x-24[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1967119\">\n<div id=\"fs-id1967120\">\n<p id=\"fs-id2429855\">Find a linear equation, that has an <em>x<\/em> intercept at (\u20134, 0) and a <em>y<\/em>-intercept at (0, \u20136), if possible.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2489441\">\n<div id=\"fs-id2489442\">\n<p id=\"fs-id2757895\">Find the slope of the line in <a class=\"autogenerated-content\" href=\"#Image_04_03_220\">(Figure)<\/a>.<\/p>\n\n<div id=\"Image_04_03_220\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"438\"]<img src=\"https:\/\/cnx.org\/resources\/250ab34d33c71d7624004041bea502b7f941ee49\/CNX_Precalc_Figure_02_04_218.jpg\" alt=\"This image is a graph of a decreasing linear function on an x, y coordinate plane. The x and y-axis range from -6 to 6. The line passes through the points (0,2) and (1,0). \" width=\"438\" height=\"437\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2136527\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2136527\"]\n<p id=\"fs-id2136527\">[latex]m=-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2429836\">\n<div id=\"fs-id2429838\">\n<p id=\"fs-id2429839\">Write an equation for line in <a class=\"autogenerated-content\" href=\"#Image_04_03_221\">(Figure)<\/a>.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"438\"]<img src=\"https:\/\/cnx.org\/resources\/415a3220bb8b54ff57fb4e6e4e0339e054952dc1\/CNX_Precalc_Figure_02_04_219.jpg\" alt=\"This image is a graph showing a decreasing linear function on an x, y coordinate plane. The x and y axis range from -6 to 6. The line passes through the points (0,-1) and (-.5,0). \" width=\"438\" height=\"437\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<div id=\"Image_04_03_221\" class=\"small\">Does <a class=\"autogenerated-content\" href=\"#Table_04_03_30\">(Figure)<\/a> represent a linear function? If so, find a linear equation that models the data.<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1795368\">\n<div id=\"fs-id1795369\">\n<table id=\"Table_04_03_30\" summary=\"The table shows two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cg(x)\u201d. The values of the first row are: -6, 0, 2, 4. The values of the second row are: 14, 32, 38, 44.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>\u20136<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>14<\/td>\n<td>32<\/td>\n<td>38<\/td>\n<td>44<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2590722\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2590722\"]\n<p id=\"fs-id2590722\">Yes,[latex]\\,y=3x+32[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2448065\">\n<div id=\"fs-id2674652\">\n<p id=\"fs-id2448066\">Does <a class=\"autogenerated-content\" href=\"#Table_04_03_31\">(Figure)<\/a> represent a linear function? If so, find a linear equation that models the data.<\/p>\n\n<table id=\"Table_04_03_31\" summary=\"The table shows two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cg(x)\u201d. The values of the first row are: 1, 3, 7, 11. The values of the second row are: 4, 9, 19, 12.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td><strong><em>g<\/em>(<em>x<\/em>)<\/strong><\/td>\n<td>4<\/td>\n<td>9<\/td>\n<td>19<\/td>\n<td>12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2448041\">\n<div id=\"fs-id2448042\">\n<p id=\"fs-id2448044\">At 6 am, an online company has sold 120 items that day. If the company sells an average of 30 items per hour for the remainder of the day, write an expression to represent the number of items that were sold[latex]\\,n\\,[\/latex]after 6 am.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2549693\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2549693\"]\n<p id=\"fs-id2549693\">[latex]y=30x+120[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2241342\">For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.<\/p>\n\n<div id=\"fs-id2241346\">\n<div id=\"fs-id2241347\">\n<p id=\"fs-id2241348\">[latex]\\begin{array}{c}y=\\frac{3}{4}x-9\\\\ -4x-3y=8\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2674691\">\n<div id=\"fs-id2674692\">\n<p id=\"fs-id2674693\">[latex]\\begin{array}{c}-2x+y=3\\\\ 3x+\\frac{3}{2}y=5\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2590764\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2590764\"]\n<p id=\"fs-id2590764\">Neither<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2590768\">\n<div id=\"fs-id2590769\">\n<p id=\"fs-id2590770\">Find the <em>x<\/em>- and <em>y<\/em>-intercepts of the equation[latex]\\,2x+7y=-14.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2489142\">\n<div id=\"fs-id2489143\">\n<p id=\"fs-id2489144\">Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is the pair of lines parallel, perpendicular, or neither?<\/p>\n<p id=\"fs-id2489148\">Line 1: Passes through[latex]\\,\\left(-2,-6\\right)\\,[\/latex]and[latex]\\,\\left(3,14\\right)[\/latex]<\/p>\n<p id=\"fs-id1505527\">Line 2: Passes through[latex]\\,\\left(2,6\\right)\\,[\/latex]and[latex]\\,\\left(4,14\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2365321\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2365321\"]\n<p id=\"fs-id2365321\">Line 1:[latex]\\,m=4;[\/latex]Line 2:[latex]\\,m=4;[\/latex]parallel<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2267122\">\n<div id=\"fs-id2267123\">\n<p id=\"fs-id2267124\">Write an equation for a line perpendicular to[latex]\\,f\\left(x\\right)=4x+3\\,[\/latex]and passing through the point[latex]\\,\\left(8,10\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2437806\">\n<div id=\"fs-id2437807\">\n<p id=\"fs-id1720532\">Sketch a line with a <em>y<\/em>-intercept of[latex]\\,\\left(0,\\text{5}\\right)\\,[\/latex]and slope[latex]\\,-\\frac{5}{2}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1801251\"]Show Solution[\/reveal-answer][hidden-answer a=\"1801251\"]<img src=\"https:\/\/cnx.org\/resources\/db97384e013f70779f231db0874722ef0d27ddf7\/CNX_Precalc_Figure_02_04_220.jpg\" alt=\"This image is a graph showing a decreasing linear function on an x, y coordinate plane. The x and y-axis range from -6 to 6. The line passes through the points (0,5) and (2,0) and a slope of: -5\/2. \">\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1801268\">\n<div id=\"fs-id1801269\">\n<p id=\"fs-id2406692\">Graph of the linear function[latex]\\,f\\left(x\\right)=-x+6.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2800116\">\n<div id=\"fs-id2800118\">\n<p id=\"fs-id2800119\">For the two linear functions, find the point of intersection:[latex]\\begin{array}{c}x=y+2\\\\ 2x-3y=-1\\end{array}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2436158\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2436158\"]\n<p id=\"fs-id2436158\">[latex]\\left(7,5\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2436183\">\n<div id=\"fs-id2436184\">\n<p id=\"fs-id2436185\">A car rental company offers two plans for renting a car.<\/p>\n<p id=\"fs-id2436188\">Plan A: $25 per day and $0.10 per mile<\/p>\n<p id=\"fs-id2436192\">Plan B: $40 per day with free unlimited mileage<\/p>\n<p id=\"fs-id2436195\">How many miles would you need to drive for plan B to save you money?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2436202\">\n<div id=\"fs-id2436203\">\n<p id=\"fs-id2436204\">Find the area of a triangle bounded by the <em>y<\/em> axis, the line[latex]\\,f\\left(x\\right)=12-4x,[\/latex]and the line perpendicular to[latex]\\,f\\,[\/latex]that passes through the origin.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2804285\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2804285\"]\n<p id=\"fs-id2804285\">16.94 square units<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2804288\">\n<div id=\"fs-id2804289\">\n<p id=\"fs-id2804290\">A town\u2019s population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2804301\">\n<div id=\"fs-id2804302\">\n<p id=\"fs-id2804303\">The number of people afflicted with the common cold in the winter months dropped steadily by 25 each year since 2002 until 2012. In 2002, 8,040 people were inflicted. Find the linear function that models the number of people afflicted with the common cold[latex]\\,C\\,[\/latex]as a function of the year,[latex]\\,t.\\,[\/latex]When will less than 6,000 people be afflicted?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2436241\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2436241\"]\n<p id=\"fs-id2436241\">2083<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"291\"]<img src=\"https:\/\/cnx.org\/resources\/bb29c461e3cdf022bfd0e468c15d2745a5280fb8\/CNX_Precalc_Figure_02_04_222.jpg\" alt=\"This image is a graph showing the company's profit from 1985 at around $15,000 to 2010 at about $32,500. The x-axis goes from 0 to 30 in intervals of 5 and the y-axis goes from 0 to 35,000 in intervals of 5,000.\" width=\"291\" height=\"285\"> <strong>Figure 14.<\/strong>[\/caption]\n<p id=\"fs-id2436244\">For the following exercises, use the graph in <a class=\"autogenerated-content\" href=\"#Figure_04_03_224\">(Figure)<\/a>, showing the profit,[latex]y,[\/latex]in thousands of dollars, of a company in a given year,[latex]\\,x,[\/latex]where[latex]\\,x\\,[\/latex]represents years since 1980.<\/p>\n\n<div id=\"fs-id2436305\">\n<div id=\"fs-id2436306\">\n<p id=\"fs-id2436307\">Find the linear function[latex]\\,y,[\/latex]where[latex]\\,y\\,[\/latex]depends on[latex]\\,x,[\/latex]the number of years since 1980.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2363849\">\n<div id=\"fs-id2363850\">\n<p id=\"fs-id2363851\">Find and interpret the <em>y<\/em>-intercept.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2363860\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2363860\"]\n<p id=\"fs-id2363860\">[latex]\\left(0,10,625\\right);[\/latex]In 1980, the profit was $10,625,000.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2387148\">\n<div id=\"fs-id2387149\">\n<p id=\"fs-id2387150\">In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly.<\/p>\n\n<ol id=\"fs-id2387155\" type=\"a\">\n \t<li>How much did the population drop between the year 2004 and 2012?<\/li>\n \t<li>What is the average population decline per year?<\/li>\n \t<li>Find an equation for the population, <em>P<\/em>, of the school <em>t<\/em> years after 2004.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id2387205\">\n<div id=\"fs-id2387206\">\n<p id=\"fs-id2387207\">Draw a scatter plot for the data provided in <a class=\"autogenerated-content\" href=\"#Table_04_03_32\">(Figure)<\/a>. Then determine whether the data appears to be linearly related.<\/p>\n\n<table id=\"Table_04_03_32\" summary=\"This table shows two rows and six columns. The values in the first row are: 0, 2, 4, 6, 8, 10. The values in the second row are: -450, -200, 10, 265, 500 and 755.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u2013450<\/td>\n<td>\u2013200<\/td>\n<td>10<\/td>\n<td>265<\/td>\n<td>500<\/td>\n<td>755<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2804389\"]Show Solution[\/reveal-answer][hidden-answer a=\"2804389\"]<img src=\"https:\/\/cnx.org\/resources\/c7a24d60ff818a189bac3a4fdd21e60511778fc7\/CNX_Precalc_Figure_02_04_231.jpg\" alt=\"Scatterplot with a collection of points: (0,-450); (2,-200); (4,10); (6,265); (8,500) and (10,755). The data appears linear.\">\n\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id2804427\">\n<div id=\"fs-id2804428\">\n<p id=\"fs-id2804429\">Draw a best-fit line for the plotted data.<img src=\"https:\/\/cnx.org\/resources\/9cf59d69cd7c32d0cd7612f75bcd0f2ed9f0b685\/CNX_Precalc_Figure_02_04_223.jpg\" alt=\"Scatterplot with domain from 2 to 10 and range from 20 from 33. The points plotted are (2,20); (4,23); (6,26); (8,26); and (10,32).\"><\/p>\nFor the following exercises, use <a class=\"autogenerated-content\" href=\"#Table_04_03_33\">(Figure)<\/a>, which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year.\n\n<\/div>\n<\/div>\n<table id=\"Table_04_03_33\" summary=\"This table shows two rows and six columns. The first row is labeled: \u201cYear\u201d and the second is labeled: \u201cPercent Graduates\u201d. The values in the first row are: 2000, 2002, 2005, 2007, 2010. The values in the second row are: 8.5, 8.0, 7.2, 6.7 and 6.4\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2000<\/td>\n<td>2002<\/td>\n<td>2005<\/td>\n<td>2007<\/td>\n<td>2010<\/td>\n<\/tr>\n<tr>\n<td><strong>Percent Graduates<\/strong><\/td>\n<td>8.5<\/td>\n<td>8.0<\/td>\n<td>7.2<\/td>\n<td>6.7<\/td>\n<td>6.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id2406470\">\n<div id=\"fs-id2406471\">\n<p id=\"fs-id2406472\">Determine whether the trend appears linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2406477\"]Show Solution[\/reveal-answer][hidden-answer a=\"2406477\"]\n\n<img src=\"https:\/\/cnx.org\/resources\/88451207483f635c2c1c476a528e3135da0992b0\/CNX_Precalc_Figure_02_04_232.jpg\" alt=\"Scatterplot with a collection of points: (2000,8.5); (2002,8); (2005,7.2); (2007,6.7); and (2010,6.4). The data appears linear\">\n<p id=\"fs-id2406495\">[latex]y=-0.219x+445.990[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2363667\">\n<div id=\"fs-id2406523\">\n<p id=\"fs-id2406524\">In what year will the percentage drop below 4%?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2406532\">\n<div id=\"fs-id2406533\">\n<p id=\"fs-id2406534\">Based on the set of data given in <a class=\"autogenerated-content\" href=\"#Table_04_03_34\">(Figure)<\/a>, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient. Round to three decimal places of accuracy.<\/p>\n\n<table id=\"Table_04_03_34\" summary=\"This table shows two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 16, 18, 20, 24, 26. The values in the second row are: 106, 110, 115, 120, 125.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>16<\/td>\n<td>18<\/td>\n<td>20<\/td>\n<td>24<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong><em>y<\/em><\/strong><\/td>\n<td>106<\/td>\n<td>110<\/td>\n<td>115<\/td>\n<td>120<\/td>\n<td>125<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2406661\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2406661\"]\n<p id=\"fs-id2406661\">[latex]y=1.820x+77.349,r=0.986[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2406691\">For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:<\/p>\n<p id=\"fs-id2406697\">[latex]\\left(4,500,2000\\right);\\left(4,700,2001\\right);\\left(5,200,2003\\right);\\left(5,800,2006\\right)[\/latex]<\/p>\n\n<div id=\"fs-id2621231\">\n<div id=\"fs-id2621232\">\n<p id=\"fs-id2621233\">Use linear regression to determine a function <em>y<\/em>, where the year depends on the population. Round to three decimal places of accuracy.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2621264\">\n<div id=\"fs-id2621265\">\n<p id=\"fs-id2621266\">Predict when the population will hit 20,000.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2621270\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2621270\"]\n<p id=\"fs-id2621270\">2070<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2621274\">\n<div id=\"fs-id2621275\">\n<p id=\"fs-id2621276\">What is the correlation coefficient for this model?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2621300\">\n \t<dt>correlation coefficient<\/dt>\n \t<dd id=\"fs-id2621304\">a value,[latex]\\,r,[\/latex]between \u20131 and 1 that indicates the degree of linear correlation of variables, or how closely a regression line fits a data set.<\/dd>\n<\/dl>\n<dl id=\"fs-id1672566\">\n \t<dt>extrapolation<\/dt>\n \t<dd id=\"fs-id1672569\">predicting a value outside the domain and range of the data<\/dd>\n<\/dl>\n<dl id=\"fs-id1672572\">\n \t<dt>interpolation<\/dt>\n \t<dd id=\"fs-id1672575\">predicting a value inside the domain and range of the data<\/dd>\n<\/dl>\n<dl id=\"fs-id1672579\">\n \t<dt>least squares regression<\/dt>\n \t<dd id=\"fs-id1672582\">a statistical technique for fitting a line to data in a way that minimizes the differences between the line and data values<\/dd>\n<\/dl>\n<dl id=\"fs-id1672586\">\n \t<dt>model breakdown<\/dt>\n \t<dd id=\"fs-id1672589\">when a model no longer applies after a certain point<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section you will:<\/p>\n<ul>\n<li>Draw and interpret scatter diagrams.<\/li>\n<li>Use a graphing utility to find the line of best fit.<\/li>\n<li>Distinguish between linear and nonlinear relations.<\/li>\n<li>Fit a regression line to a set of data and use the linear model to make predictions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id2164914\">A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.<\/p>\n<div id=\"fs-id2188781\" class=\"bc-section section\">\n<h3>Drawing and Interpreting Scatter Plots<\/h3>\n<p id=\"fs-id1618160\">A <span class=\"no-emphasis\">scatter plot<\/span> is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a <span class=\"no-emphasis\">linear model<\/span>, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions. <a class=\"autogenerated-content\" href=\"#Figure_04_03_001\">(Figure)<\/a> shows a sample scatter plot.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/d85e87d6a305be35528f5ab89b201056fdbe8d53\/CNX_Precalc_Figure_02_04_001.jpg\" alt=\"Scatter plot, titled 'Final Exam Score VS Age'. The x-axis is the age, and the y-axis is the final exam score. The range of ages are between 20s - 50s, and the range for scores are between upper 50s and 90s.\" width=\"487\" height=\"337\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>A scatter plot of age and final exam score variables<\/figcaption><\/figure>\n<p id=\"fs-id1621860\">Notice this scatter plot does <em>not<\/em> indicate a <span class=\"no-emphasis\">linear relationship<\/span>. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.<\/p>\n<div id=\"Example_04_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1802821\">\n<div id=\"fs-id1798376\">\n<h3>Using a Scatter Plot to Investigate Cricket Chirps<\/h3>\n<p id=\"fs-id1797122\"><a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a> shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit<a class=\"footnote\" title=\"Selected data from http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/. Retrieved Aug 3, 2010\" id=\"return-footnote-74-1\" href=\"#footnote-74-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. Plot this data, and determine whether the data appears to be linearly related.<\/p>\n<table id=\"Table_04_03_01\" summary=\"Table with two rows and ten columns. The first column is labeled: \u201cChirps\u201d and the second is labeled: \u201cTemperature\u201d. The values for chirps are: 44, 35, 20.4, 33, 31, 35, 18.5, 37, 26. The values for Temperature are: 80.5, 70.5, 57, 66, 68, 72, 52, 73.5, 53.\">\n<caption>Cricket Chirps vs Air Temperature<\/caption>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Plotting this data, as depicted in <a class=\"autogenerated-content\" href=\"#Figure_04_03_002\">(Figure)<\/a> suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/0bd821366e09c7df8ba83f57a8b07d4aa8fb15aa\/CNX_Precalc_Figure_02_04_002.jpg\" alt=\"Scatter plot, titled 'Cricket Chirps vs. Air Temperature'. The x-axis is the Cricket Chirps in 15 Seconds, and the y-axis is the Temperature (F). The line regression is generally positive.\" width=\"487\" height=\"386\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2672477\" class=\"bc-section section\">\n<h3>Finding the Line of Best Fit<\/h3>\n<p id=\"fs-id1798587\">Once we recognize a need for a linear function to model that data, the natural follow-up question is \u201cwhat is that linear function?\u201d One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the <em>y<\/em>-intercept. We can approximate the slope of the line by extending it until we can estimate the[latex]\\,\\frac{\\text{rise}}{\\text{run}}.[\/latex]<\/p>\n<div id=\"Example_04_03_02\" class=\"textbox examples\">\n<div id=\"fs-id2262247\">\n<div id=\"fs-id1575271\">\n<h3>Finding a Line of Best Fit<\/h3>\n<p id=\"fs-id2268675\">Find a linear function that fits the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a> by \u201ceyeballing\u201d a line that seems to fit.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1497780\">On a graph, we could try sketching a line. Using the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of<\/p>\n<div id=\"fs-id1801316\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}m=\\frac{60}{50}\\hfill \\\\ \\,\\,\\,\\,=1.2\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2651822\">and a <em>y<\/em>-intercept at 30. This gives an equation of<\/p>\n<div id=\"fs-id1676754\" class=\"unnumbered aligncenter\">[latex]T\\left(c\\right)=1.2c+30[\/latex]<\/div>\n<p>where[latex]\\,c\\,[\/latex]is the number of chirps in 15 seconds, and[latex]\\,T\\left(c\\right)\\,[\/latex]is the temperature in degrees Fahrenheit. The resulting equation is represented in <a class=\"autogenerated-content\" href=\"#Figure_04_03_003\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/c3b09ea16d20f7e7e4ac733ef5fade8c17a5d929\/CNX_Precalc_Figure_02_04_003.jpg\" alt=\"Scatter plot, showing the line of best fit: T(c) = 1.2c + 30. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"432\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<div id=\"fs-id1486842\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1720506\">This linear equation can then be used to approximate answers to various questions we might ask about the trend.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1720372\" class=\"bc-section section\">\n<h4>Recognizing Interpolation or Extrapolation<\/h4>\n<p id=\"fs-id1711565\">While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as <strong>interpolation <\/strong>when we predict a value inside the domain and range of the data. The process of <strong>extrapolation <\/strong>is used when we predict a value outside the domain and range of the data.<\/p>\n<p id=\"fs-id2208869\"><a class=\"autogenerated-content\" href=\"#Figure_04_03_004\">(Figure)<\/a> compares the two processes for the cricket-chirp data addressed in <a class=\"autogenerated-content\" href=\"#Example_04_03_02\">(Figure)<\/a>. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.<\/p>\n<p id=\"fs-id2560478\">There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when[latex]\\,x=50,[\/latex]that is in 50 years, the model would not apply because we could not account for factors fifty years in the future.<\/p>\n<div id=\"Figure_04_03_004\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/fd26d5773978743dded07a498c069ea62e7cc5b1\/CNX_Precalc_Figure_02_04_004.jpg\" alt=\"Scatter plot, showing the line of best fit. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'. The area around the scattered points is enclosed in a box labeled: Interpolation. The area outside of this box is labeled: Extrapolation.\" width=\"487\" height=\"430\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4. <\/strong>Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.<\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id2570361\" class=\"textbox key-takeaways\">\n<h3>Interpolation and Extrapolation<\/h3>\n<p id=\"fs-id1633393\">Different methods of making predictions are used to analyze data.<\/p>\n<p id=\"eip-872\">The method of interpolation involves predicting a value inside the domain and\/or range of the data.<\/p>\n<p>The method of extrapolation involves predicting a value outside the domain and\/or range of the data.<\/p>\n<p>Model breakdown occurs at the point when the model no longer applies.<\/p>\n<\/div>\n<div id=\"Example_04_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1691726\">\n<div id=\"fs-id1620023\">\n<h3>Understanding Interpolation and Extrapolation<\/h3>\n<p id=\"fs-id1842101\">Use the cricket data from <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a> to answer the following questions:<\/p>\n<ol id=\"fs-id1713857\" type=\"a\">\n<li>Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\n<li>Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make the prediction, and discuss whether it is reasonable.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1570270\" type=\"a\">\n<li>The number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is inside the domain of our data, so would be interpolation. Using our model:\n<div id=\"fs-id1202174\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\text{T (30)}& =& \\text{30 + 1.2(30)}\\hfill \\\\ & =& \\text{66 degrees}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1555289\">Based on the data we have, this value seems reasonable.<\/p>\n<\/li>\n<li>The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation because 40 is outside the range of our data. Using our model:\n<div id=\"fs-id1251848\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}40=30+1.2c\\hfill \\\\ 10=1.2c\\hfill \\\\ \\text{ }c\\approx 8.33\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p>We can compare the regions of interpolation and extrapolation using <a class=\"autogenerated-content\" href=\"#Figure_04_03_005\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 485px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/3bb7ccc3fc050a07015bdc1ea2861e402262b9fe\/CNX_Precalc_Figure_02_04_005.jpg\" alt=\"Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps vs. Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'. An additional point is plotted inside of the box to represent an interpolated point. There is another additional point plotted outside of the box to represent an extrapolated point.\" width=\"485\" height=\"429\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<div id=\"fs-id1422108\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1768136\">Our model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether below around 50 degrees.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2181223\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_01\">\n<div id=\"fs-id1589415\">\n<p id=\"fs-id1585053\">According to the data from <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a>, what temperature can we predict it is if we counted 20 chirps in 15 seconds?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1685205\">[latex]54\u00b0\\text{F}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Finding the Line of Best Fit Using a Graphing Utility<\/h4>\n<p id=\"fs-id2293766\">While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values<a class=\"footnote\" title=\"Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.\" id=\"return-footnote-74-2\" href=\"#footnote-74-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> . One such technique is called least squares regression and can be computed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculators<a class=\"footnote\" title=\"For example, http:\/\/www.shodor.org\/unchem\/math\/lls\/leastsq.html\" id=\"return-footnote-74-3\" href=\"#footnote-74-3\" aria-label=\"Footnote 3\"><sup class=\"footnote\">[3]<\/sup><\/a> <sup id=\"footnote-ref3\"><\/sup>. Least squares regression is one means to determine the line that best fits the data, and here we will refer to this method as linear regression.<\/p>\n<div id=\"fs-id2627255\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id2251507\"><strong>Given data of input and corresponding outputs from a linear function, find the best fit line using linear regression.<\/strong><\/p>\n<ol id=\"fs-id1503997\" type=\"1\">\n<li>Enter the input in List 1 (L1).<\/li>\n<li>Enter the output in List 2 (L2).<\/li>\n<li>On a graphing utility, select Linear Regression (LinReg).<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_03_04\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1632771\">\n<h3>Finding a Least Squares Regression Line<\/h3>\n<p id=\"fs-id2575012\">Find the least squares <span class=\"no-emphasis\">regression line<\/span> using the cricket-chirp data in <a class=\"autogenerated-content\" href=\"#Table_04_03_02\">(Figure)<\/a>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1587366\" type=\"1\">\n<li>Enter the input (chirps) in List 1 (L1).<\/li>\n<li>Enter the output (temperature) in List 2 (L2). See <a class=\"autogenerated-content\" href=\"#Table_04_03_02\">(Figure)<\/a>.<br \/>\n<table id=\"Table_04_03_02\" summary=\"This table has two rows and ten columns. The first row is labeled: \u201cL1\u201d and the second is labeled: \u201cL2\u201d. The values in the first row are: 44, 35, 20.4, 33, 31, 35, 18.5, 37, 26. The values in the second row are: 80.5, 70.5, 57, 66, 68, 72, 52, 73.5, 53.\">\n<tbody>\n<tr>\n<td><strong>L1<\/strong><\/td>\n<td>44<\/td>\n<td>35<\/td>\n<td>20.4<\/td>\n<td>33<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<td>18.5<\/td>\n<td>37<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong>L2<\/strong><\/td>\n<td>80.5<\/td>\n<td>70.5<\/td>\n<td>57<\/td>\n<td>66<\/td>\n<td>68<\/td>\n<td>72<\/td>\n<td>52<\/td>\n<td>73.5<\/td>\n<td>53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>On a graphing utility, select Linear Regression (LinReg). Using the cricket chirp data from earlier, with technology we obtain the equation:<\/li>\n<\/ol>\n<div id=\"fs-id1618400\" class=\"unnumbered aligncenter\">[latex]T\\left(c\\right)=30.281+1.143c[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1594310\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1695711\">Notice that this line is quite similar to the equation we \u201ceyeballed\u201d but should fit the data better. Notice also that using this equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees to:<\/p>\n<div id=\"fs-id2136214\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}T\\left(30\\right)=30.281+1.143\\left(30\\right)\\hfill \\\\ \\text{ }=64.571\\hfill \\\\ \\text{ }\\approx 64.6\\text{ degrees}\\hfill \\end{array}[\/latex]<\/div>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/518bc40eb0e1068ac46e62e9f2a414854c98e0f6\/CNX_Precalc_Figure_02_04_006.jpg\" alt=\"Scatter plot, showing the line of best fit: T(c) = 30.281 + 1.143c. It is titled 'Cricket Chirps vs. Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.\" width=\"487\" height=\"408\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p>The graph of the scatter plot with the least squares regression line is shown in <a class=\"autogenerated-content\" href=\"#Figure_04_03_006\">(Figure)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1592598\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id2181364\"><strong>Will there ever be a case where two different lines will serve as the best fit for the data? <\/strong><\/p>\n<p id=\"fs-id1688806\"><em>No. There is only one best fit line.<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1459299\" class=\"bc-section section\">\n<h3>Distinguishing Between Linear and Nonlinear Models<\/h3>\n<p id=\"fs-id1586029\">As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <span class=\"no-emphasis\">correlation coefficient<\/span>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a \u201ddiagnostic on\u201d selection to find the correlation coefficient, which mathematicians label as[latex]\\,r\\,[\/latex]The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.<\/p>\n<p id=\"fs-id2674051\">We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of[latex]\\,r\\,[\/latex]and the graph of the data, <a class=\"autogenerated-content\" href=\"#Figure_04_03_007\">(Figure)<\/a> shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.<\/p>\n<div id=\"Figure_04_03_007\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/f97e72fcb8b9233a23aeebdbb15e21f80163163c\/CNX_Precalc_Figure_02_04_007.jpg\" alt=\"Correlation coefficients values range from -1.0 - 1.0. Collections of dots representing an example of each kind of correlation coefficient are plotted underneath them. The closer to 1.0 the more the points are grouped tightly to form a line in the positive direction. The closer to -1.0 the more the points are grouped tightly to form a line in the negative direction. The closer to 0 the points are very scattered and do not form a line. Several shapes are displayed at the bottom row, none of which are lines, but all of them have values of 0.\" width=\"975\" height=\"434\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong> Plotted data and related correlation coefficients. (credit: \u201cDenisBoigelot,\u201d Wikimedia Commons)<\/figcaption><\/figure>\n<h3>Correlation Coefficient<\/h3>\n<div id=\"fs-id1415360\">\n<p id=\"fs-id1589701\">The <strong>correlation coefficient<\/strong> is a value,[latex]\\,r,[\/latex]between \u20131 and 1.<\/p>\n<ul id=\"fs-id1963658\">\n<li>[latex]r>0\\,[\/latex]suggests a positive (increasing) relationship<\/li>\n<li>[latex]r<0\\,[\/latex]suggests a negative (decreasing) relationship<\/li>\n<li>The closer the value is to 0, the more scattered the data.<\/li>\n<li>The closer the value is to 1 or \u20131, the less scattered the data is.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1367501\">\n<div id=\"fs-id1703090\">\n<h3>Finding a Correlation Coefficient<\/h3>\n<p id=\"fs-id2165511\">Calculate the correlation coefficient for cricket-chirp data in <a class=\"autogenerated-content\" href=\"#Table_04_03_01\">(Figure)<\/a>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2164982\">Because the data appear to follow a linear pattern, we can use technology to calculate[latex]\\,r\\,[\/latex]Enter the inputs and corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation coefficient,[latex]\\,r=0.9509.\\,[\/latex]This value is very close to 1, which suggests a strong increasing linear relationship.<\/p>\n<p id=\"fs-id1154034\">Note: For some calculators, the Diagnostics must be turned &#8220;on&#8221; in order to get the correlation coefficient when linear regression is performed: [2nd]&gt;[0]&gt;[alpha][x\u20131], then scroll to DIAGNOSTICSON.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1840824\" class=\"bc-section section\">\n<h3>Fitting a Regression Line to a Set of Data<\/h3>\n<p id=\"fs-id1685569\">Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.<\/p>\n<div id=\"Example_04_03_06\" class=\"textbox examples\">\n<div id=\"fs-id917127\">\n<div id=\"fs-id1827875\">\n<h3>Using a Regression Line to Make Predictions<\/h3>\n<p id=\"fs-id1587319\">Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in <a class=\"autogenerated-content\" href=\"#Table_04_03_03\">(Figure)<\/a>.<a class=\"footnote\" title=\"http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html\" id=\"return-footnote-74-4\" href=\"#footnote-74-4\" aria-label=\"Footnote 4\"><sup class=\"footnote\">[4]<\/sup><\/a><sup id=\"footnote-ref4\"><\/sup> Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.<\/p>\n<table id=\"Table_04_03_03\" summary=\"This table has two rows and twelve columns. The first row is labeled: \u201cYear\u201d and the second is labeled: \u201cConsumption (billions of gallons)\u201d. The values in the first row are: \u201994, \u201995, \u201996, \u201997, \u201998, \u201999, \u201900, \u201901, \u201902, \u201903, \u201904. The values in the second row are: 113, 116, 118, 119, 123, 125, 126, 128, 131, 133, 136.\">\n<tbody>\n<tr>\n<td style=\"width: 154px\"><strong>Year<\/strong><\/td>\n<td style=\"width: 31px\">&#8217;94<\/td>\n<td style=\"width: 31px\">&#8217;95<\/td>\n<td style=\"width: 31px\">&#8217;96<\/td>\n<td style=\"width: 31px\">&#8217;97<\/td>\n<td style=\"width: 31px\">&#8217;98<\/td>\n<td style=\"width: 31px\">&#8217;99<\/td>\n<td style=\"width: 31px\">&#8217;00<\/td>\n<td style=\"width: 31px\">&#8217;01<\/td>\n<td style=\"width: 31px\">&#8217;02<\/td>\n<td style=\"width: 31px\">&#8217;03<\/td>\n<td style=\"width: 31px\">&#8217;04<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 154px\"><strong>Consumption (billions of gallons)<\/strong><\/td>\n<td style=\"width: 31px\">113<\/td>\n<td style=\"width: 31px\">116<\/td>\n<td style=\"width: 31px\">118<\/td>\n<td style=\"width: 31px\">119<\/td>\n<td style=\"width: 31px\">123<\/td>\n<td style=\"width: 31px\">125<\/td>\n<td style=\"width: 31px\">126<\/td>\n<td style=\"width: 31px\">128<\/td>\n<td style=\"width: 31px\">131<\/td>\n<td style=\"width: 31px\">133<\/td>\n<td style=\"width: 31px\">136<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The scatter plot of the data, including the least squares regression line, is shown in <a class=\"autogenerated-content\" href=\"#Figure_04_03_008\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/46361ea50eb0fd022efd04ebd947b65218bcc6f8\/CNX_Precalc_Figure_02_04_008.jpg\" alt=\"Scatter plot, showing the line of best fit. It is titled 'Gas Consumption VS Year'. The x-axis is 'Year After 1994', and the y-axis is 'Gas Consumption (billions of gallons)'. The points are strongly positively correlated and the line of best fit goes through most of the points completely.\" width=\"487\" height=\"384\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1588946\">We can introduce new input variable,[latex]\\,t,[\/latex]representing years since 1994.<\/p>\n<p id=\"fs-id1565498\">The least squares regression equation is:<\/p>\n<div id=\"fs-id2065061\" class=\"unnumbered aligncenter\">[latex]C\\left(t\\right)=113.318+2.209t[\/latex]<\/div>\n<p id=\"fs-id1503420\">Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.<\/p>\n<p id=\"fs-id2786561\">Using this to predict consumption in 2008[latex]\\,\\left(t=14\\right),[\/latex]<\/p>\n<div id=\"fs-id1737282\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}C\\left(14\\right)=113.318+2.209\\left(14\\right)\\hfill \\\\ \\text{ }=144.244\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1716786\">The model predicts 144.244 billion gallons of gasoline consumption in 2008.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1799110\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_03_02\">\n<div id=\"fs-id1549586\">\n<p id=\"fs-id1619902\">Use the model we created using technology in <a class=\"autogenerated-content\" href=\"#Example_04_03_06\">(Figure)<\/a> to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1608557\">150.871 billion gallons; extrapolation<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1551115\" class=\"precalculus media\">\n<p id=\"fs-id1582076\">Access these online resources for additional instruction and practice with fitting linear models to data.<\/p>\n<ul id=\"bulleted\">\n<li><a href=\"http:\/\/Openstaxcollege.org\/l\/introregress\">Introduction to Regression Analysis<\/a><\/li>\n<li><a href=\"http:\/\/Openstaxcollege.org\/l\/linearregress\">Linear Regression<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1486512\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1862819\">\n<li>Scatter plots show the relationship between two sets of data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_01\">(Figure)<\/a>.<\/li>\n<li>Scatter plots may represent linear or non-linear models.<\/li>\n<li>The line of best fit may be estimated or calculated, using a calculator or statistical software. See <a class=\"autogenerated-content\" href=\"#Example_04_03_02\">(Figure)<\/a>.<\/li>\n<li>Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_03\">(Figure)<\/a>.<\/li>\n<li>The correlation coefficient,[latex]\\,r,[\/latex]indicates the degree of linear relationship between data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_04\">(Figure)<\/a>.<\/li>\n<li>A regression line best fits the data. See <a class=\"autogenerated-content\" href=\"#Example_04_03_05\">(Figure)<\/a>.<\/li>\n<li>The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See <a class=\"autogenerated-content\" href=\"#Example_04_03_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id942179\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2598318\">\n<div id=\"fs-id2248686\">\n<p id=\"fs-id1598256\">Describe what it means if there is a model breakdown when using a linear model.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1412205\">When our model no longer applies, after some value in the domain, the model itself doesn\u2019t hold.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2638023\">\n<div id=\"fs-id1690731\">\n<p id=\"fs-id2090338\">What is interpolation when using a linear model?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id888239\">\n<div id=\"fs-id940827\">\n<p id=\"fs-id1709982\">What is extrapolation when using a linear model?<\/p>\n<\/div>\n<div id=\"fs-id926962\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2113896\">We predict a value outside the domain and range of the data.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id918008\">\n<div id=\"fs-id1472127\">\n<p id=\"fs-id894113\">Explain the difference between a positive and a negative correlation coefficient.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2158268\">\n<div>\n<p id=\"fs-id1791770\">Explain how to interpret the absolute value of a correlation coefficient.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1570206\">The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1619206\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id2484796\">\n<div id=\"fs-id1540854\">\n<p id=\"fs-id2552133\">A regression was run to determine whether there is a relationship between hours of TV watched per day[latex]\\,\\left(x\\right)\\,[\/latex]and number of sit-ups a person can do[latex]\\,\\left(y\\right).\\,[\/latex]The results of the regression are given below. Use this to predict the number of sit-ups a person who watches 11 hours of TV can do.<\/p>\n<div id=\"fs-id1487852\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=ax+b\\hfill \\\\ a=-1.341\\hfill \\\\ b=32.234\\hfill \\\\ \\text{ }r=-0.896\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1541363\">\n<div id=\"fs-id2643682\">\n<p id=\"fs-id1626298\">A regression was run to determine whether there is a relationship between the diameter of a tree ([latex]\\,x,[\/latex]in inches) and the tree\u2019s age ([latex]\\,y,[\/latex]in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches.<\/p>\n<div id=\"fs-id1474007\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=ax+b\\hfill \\\\ a=6.301\\hfill \\\\ b=-1.044\\hfill \\\\ \\text{ }r=-0.970\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2159539\">61.966 years<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2051997\">For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?<\/p>\n<div id=\"fs-id1790165\">\n<div id=\"fs-id2634740\">\n<table id=\"Table_04_03_04\" class=\"unnumbered\" summary=\"This table includes two rows and six columns. The values in the first row are 0, 2, 4, 6, 8, 10. The values in the second row are: -22, -19, -15, -11, -6, -2.\">\n<tbody>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u201322<\/td>\n<td>\u201319<\/td>\n<td>\u201315<\/td>\n<td>\u201311<\/td>\n<td>\u20136<\/td>\n<td>\u20132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2217614\">\n<div id=\"fs-id2097711\">\n<table id=\"Table_04_03_35\" class=\"unnumbered\" summary=\"This table has two rows and six columns. The values in the first row are: 1, 2, 3, 4, 5, 6. The values in the second row are 46, 50, 59, 75, 100, 136.\">\n<tbody>\n<tr>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>46<\/td>\n<td>50<\/td>\n<td>59<\/td>\n<td>75<\/td>\n<td>100<\/td>\n<td>136<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/7830e8f085112456858cc05c30f6dddc829d133f\/CNX_Precalc_Figure_02_04_234.jpg\" alt=\"Scatter plot with a collection of points appearing at (1,46); (2,50); (3,59); (4,75); (5, 100); and (6,136); they do not appear linear\" \/><\/p>\n<p id=\"fs-id1534066\">No.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1541935\">\n<div id=\"fs-id1374381\">\n<table id=\"Table_04_03_05\" class=\"unnumbered\" summary=\"Two rows and six columns. The values in the first row are: 100, 250, 300, 450, 600, 750. The values in the second row are: 12, 12.6, 13.1, 14, 14.5, 15.2.\">\n<tbody>\n<tr>\n<td>100<\/td>\n<td>250<\/td>\n<td>300<\/td>\n<td>450<\/td>\n<td>600<\/td>\n<td>750<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>12.6<\/td>\n<td>13.1<\/td>\n<td>14<\/td>\n<td>14.5<\/td>\n<td>15.2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id894444\">\n<div id=\"fs-id893652\">\n<table id=\"Table_04_03_06\" class=\"unnumbered\" summary=\"Two rows and six columns. The values in the first row are: 1, 3, 5, 7, 9, 11. The values in the second row are 1, 9, 28, 65, 125, 216.\">\n<tbody>\n<tr>\n<td>1<\/td>\n<td>3<\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>9<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>9<\/td>\n<td>28<\/td>\n<td>65<\/td>\n<td>125<\/td>\n<td>216<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/aec19f39026dfb8f3d1266ac042c3bb70e4c730e\/CNX_Precalc_Figure_02_04_236.jpg\" alt=\"Scatterplot with a collection of points at (1,1); (3,9); (5,28); (7,65); (9,125); and (11,216); they do not appear linear\" \/><\/p>\n<p id=\"fs-id1542149\">No.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1549099\">\n<div id=\"fs-id2257504\">\n<p id=\"fs-id1536153\">For the following data, draw a scatter plot. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation? Eyeball the line, and estimate the answer.<\/p>\n<table id=\"Table_04_03_07\" class=\"unnumbered\" summary=\"This table includes two columns and six rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cPopulation\u201d. The values in the first column are: 1990, 1995, 2000, 2005, 2010. The values in the second column are 11,500; 12,100; 12,700; 13,000; 13,750.\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Population<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>11,500<\/td>\n<\/tr>\n<tr>\n<td>1995<\/td>\n<td>12,100<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>12,700<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>13,000<\/td>\n<\/tr>\n<tr>\n<td>2010<\/td>\n<td>13,750<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2202140\">\n<div id=\"fs-id1737923\">\n<p id=\"fs-id1604883\">For the following data, draw a scatter plot. If we wanted to know when the temperature would reach 28\u00b0F, would the answer involve interpolation or extrapolation? Eyeball the line and estimate the answer.<\/p>\n<table id=\"Table_04_03_08\" class=\"unnumbered\" summary=\"This table includes two rows and six columns. The first column is labeled: \u201cTemperature, \u00b0F\u201d and the second is labeled: \u201cTime, seconds\u201d. The values in the first row are 16, 18, 20, 25, 30. The values in the second row are: 46, 50, 54, 55, 62.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>Temperature,\u00b0F<\/strong><\/td>\n<td>16<\/td>\n<td>18<\/td>\n<td>20<\/td>\n<td>25<\/td>\n<td>30<\/td>\n<\/tr>\n<tr>\n<td><strong>Time, seconds<\/strong><\/td>\n<td>46<\/td>\n<td>50<\/td>\n<td>54<\/td>\n<td>55<\/td>\n<td>62<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/f57338a3b93fc667a821d10879a31024671db499\/CNX_Precalc_Figure_02_04_238.jpg\" alt=\"Scatterplot with a collection of points at (16,46); (18,50); (20,54); (25,55); and (30,62); they appear nonlinear\" \/><\/p>\n<p id=\"fs-id1470010\">Interpolation. About [latex]\\,60\u00b0F.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1475649\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<figure style=\"width: 465px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/027e806014d4c7ede73d26b885a3951f322d105e\/CNX_Precalc_Figure_02_04_204ab.jpg\" alt=\"Side-by-side scatter plots. The first is a scattered correlation in the positive direction. The second is a scattered correlation in the negative direction\" width=\"465\" height=\"864\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<figure style=\"width: 465px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/91207bda2341bb80d0e79a83f80225aa5b14414c\/CNX_Precalc_Figure_02_04_204cd.jpg\" alt=\"Side-by-side scatter plots. The first has a strong negative correlation with all the points spaced out evenly near the top and center, but more spread out near the bottom. The second has a strong positive correlation, with the points more spread out near the bottom and closer together near the center and top.\" width=\"465\" height=\"864\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id2570326\">For the following exercises, match each scatterplot with one of the four specified correlations in <a class=\"autogenerated-content\" href=\"#Figure_04_03_201\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Figure_04_03_202a\">(Figure)<\/a><strong>.<\/strong><span id=\"fs-id1566729\"><\/span><\/p>\n<div id=\"fs-id1284066\">\n<div>\n<p id=\"fs-id1380305\">[latex]r=0.\\text{95}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1619998\">\n<div id=\"fs-id834228\">\n<p id=\"fs-id2477405\">[latex]r=-0.\\text{89}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1395980\">[latex]\\text{This value of r indicates a strong negative correlation or slope, so C}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2281428\">\n<div id=\"fs-id1501585\">\n<p id=\"fs-id1738451\">[latex]r=-0.26[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2660448\">\n<div id=\"fs-id1412044\">[latex]r=-0.39[\/latex]<\/div>\n<div id=\"fs-id995908\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1708754\">[latex]\\text{This value of r indicates a weak negative correlation, so B}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1691868\">For the following exercises, draw a best-fit line for the plotted data.<\/p>\n<div id=\"fs-id1374546\">\n<div id=\"fs-id1694634\"><span id=\"fs-id2094007\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/6cb41d21268059aa5b04cfcfbb4f1d62fb544070\/CNX_Precalc_02_04_202.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of 4 to 9. The points are at (0,5); (2.1,4.2); (3.5,6); (4.5,6.5); (5.5,6.8); (7,7.4); (8,8.5); (9,8); and (10,9).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id2366654\">\n<div id=\"fs-id2366655\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/e17241eb73231fa5f6f7a11d8f02329400609555\/CNX_Precalc_Figure_02_04_204.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of -1 to 4. The points are at (0,1.5); (1.5, -0.1); (2.1,1.9); (3.4, 1.5); (4.5,2.5); (5.8,2.2); (6.8,3.8); (7.8,3.6); (8.8,2); and (10,2.4).\" \/><\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/71e838bf549242004a7f1eecfeeb66cde279b324\/CNX_Precalc_Figure_02_04_205.jpg\" alt=\"Scatter plot with domain 0 to 10 and a range from -1 to 4 with the line of best fit drawn going through the points: (0,1.5); (1.5, -0.1); (2.1,1.9); (3.4, 1.5); (4.5,2.5); (5.8,2.2); (6.8,3.8); (7.8,3.6); (8.8,2); and (10,2.4).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1703008\">\n<div id=\"fs-id1703009\"><span id=\"fs-id1630678\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/f8060b2872a3c3ed4a857f5e8d1113745de760bb\/CNX_Precalc_Figure_02_04_206.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and range of 0 to 7 with the points: (0,7.3); (1,7); (2.2,6); (3.6,7); (4.8,6.2); (5.8,4); (6.6,3.8); (7.9,2.4); (8.8,2); and (10,0.1).\" \/><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/4fe282978ff5bfb9bde48a0492a2e094a3d63d50\/CNX_Precalc_Figure_02_04_208.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of 2 to 6 with the points: (0,2.1); (1,3.9); (2.1,3.6); (3.6,3.9); (4.4,4); (5.6,4.2); (6.8,5); (7.8,5); (9,5.6); and (10,6).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1581440\">\n<div id=\"fs-id1581441\"><span id=\"fs-id1451072\">&nbsp;<\/span><\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/cbd36cc651fdbd957dfcd9a64989a9f9053ea0a6\/CNX_Precalc_Figure_02_04_210.jpg\" alt=\"Scatter plot with a domain of 0 to 10 and a range of 2 to 6 and the line of best fit going through the points: (0,2.1); (1,3.9); (2.1,3.6); (3.6,3.9); (4.4,4); (5.6,4.2); (6.8,5); (7.8,5); (9,5.6); and (10,6)\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2464699\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<div id=\"fs-id1690887\">\n<div id=\"fs-id1690888\">\n<p id=\"fs-id1695773\">The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given in <a class=\"autogenerated-content\" href=\"#Table_04_03_11\">(Figure)<\/a>.<a class=\"footnote\" title=\"Based on data from http:\/\/www.census.gov\/hhes\/socdemo\/education\/data\/cps\/historical\/index.html. Accessed 5\/1\/2014.\" id=\"return-footnote-74-5\" href=\"#footnote-74-5\" aria-label=\"Footnote 5\"><sup class=\"footnote\">[5]<\/sup><\/a><sup id=\"footnote-ref5\"><\/sup> Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the percentage exceed 35%?<\/p>\n<table id=\"Table_04_03_11\" summary=\"This table includes two columns and eleven rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cPercent Graduates\u201d. The values in the first column are: 1990, 1992, 1994, 1996, 1998, 2000, 2002, 2004, 2006, 2008. The values in the second column are: 21.3, 21.4, 22.2, 23.6, 24.4, 25.6, 26.7, 27.7, 28, 29.4\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Percent Graduates<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>21.3<\/td>\n<\/tr>\n<tr>\n<td>1992<\/td>\n<td>21.4<\/td>\n<\/tr>\n<tr>\n<td>1994<\/td>\n<td>22.2<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>23.6<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>24.4<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>25.6<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>26.7<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>27.7<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>28<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>29.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1798367\">\n<div id=\"fs-id1565618\">\n<p id=\"fs-id1565619\">The U.S. import of wine (in hectoliters) for several years is given in <a class=\"autogenerated-content\" href=\"#Table_04_03_12\">(Figure)<\/a>. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will imports exceed 12,000 hectoliters?<\/p>\n<table id=\"Table_04_03_12\" summary=\"This table includes two columns and eleven rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cImports\u201d. The values in the first column are: 1992, 1994, 1996, 1998, 2000, 2002, 2004, 2006, 2008, 2009. The values in the second solumn are: 2665, 2688, 3565, 4129, 4584, 5655, 6549, 7950, 8487, 9462\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Imports<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1992<\/td>\n<td>2665<\/td>\n<\/tr>\n<tr>\n<td>1994<\/td>\n<td>2688<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>3565<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>4129<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>4584<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>5655<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>6549<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>7950<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>8487<\/td>\n<\/tr>\n<tr>\n<td>2009<\/td>\n<td>9462<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1629450\">Yes, trend appears linear because[latex]\\,r=0.\\text{985}\\,[\/latex]and will exceed 12,000 near midyear, 2016, 24.6 years since 1992.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1487111\">\n<div id=\"fs-id1900537\">\n<p id=\"fs-id1900538\"><a class=\"autogenerated-content\" href=\"#Table_04_03_13\">(Figure)<\/a> shows the year and the number of people unemployed in a particular city for several years. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the number of unemployed reach 5?<\/p>\n<table id=\"Table_04_03_13\" summary=\"This table includes two columns and eleven rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cNumber Unemployed\u201d. The values in the first column are: 1990, 1992, 1994, 1996, 1998, 2000, 2002, 2004, 2006, 2008. The values in the second column are: 750, 670, 650, 605, 550, 510, 460, 420, 380, 320.\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Number Unemployed<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>750<\/td>\n<\/tr>\n<tr>\n<td>1992<\/td>\n<td>670<\/td>\n<\/tr>\n<tr>\n<td>1994<\/td>\n<td>650<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>605<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>550<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>510<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>460<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>420<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>380<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>320<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1390394\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id2368077\">For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.<\/p>\n<div id=\"fs-id1712789\">\n<div id=\"fs-id2041879\">\n<table id=\"Table_04_03_14\" class=\"unnumbered\" summary=\"This table includes two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 8, 15, 26, 31, 56. The values in the second row are: 23, 41, 53, 72, 103.\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>15<\/td>\n<td>26<\/td>\n<td>31<\/td>\n<td>56<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>23<\/td>\n<td>41<\/td>\n<td>53<\/td>\n<td>72<\/td>\n<td>103<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1689902\">[latex]y=\\text{1}.\\text{64}0x+\\text{13}.\\text{8}00,[\/latex][latex]r=0.\\text{987}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1595978\">\n<div id=\"fs-id1595979\">\n<table id=\"Table_04_03_15\" class=\"unnumbered\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 5, 7, 10, 12, 15. The values in the second row are: 4, 12, 17, 22, 24\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>10<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>4<\/td>\n<td>12<\/td>\n<td>17<\/td>\n<td>22<\/td>\n<td>24<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1474083\">\n<div id=\"fs-id1429608\">\n<table id=\"Table_04_03_16\" class=\"unnumbered\" summary=\"This table has two columns and sixteen rows. The first column is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first column are: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18. The values in the second column are: 21.9, 22.22, 22.74, 22.26, 20.78, 17.6, 16.52, 18.54, 15.76, 13.68, 14.1, 14.02, 11.94, 12.76, 11.28, 9.1.\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>3<\/td>\n<td>21.9<\/td>\n<td>10<\/td>\n<td>18.54<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>22.22<\/td>\n<td>11<\/td>\n<td>15.76<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>22.74<\/td>\n<td>12<\/td>\n<td>13.68<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>22.26<\/td>\n<td>13<\/td>\n<td>14.1<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>20.78<\/td>\n<td>14<\/td>\n<td>14.02<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>17.6<\/td>\n<td>15<\/td>\n<td>11.94<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>16.52<\/td>\n<td>16<\/td>\n<td>12.76<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1843470\">[latex]y=-0.962x+26.86, r=-0.965[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1862833\">\n<div id=\"fs-id1862834\">\n<table id=\"Table_04_03_17\" class=\"unnumbered\" summary=\"This table has two columns and eleven rows. The first column is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first column are: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. The values in the second column are: 44.8, 43.1, 38.8, 39, 38, 32.7, 30.1, 29.3, 27, 25.8\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>4<\/td>\n<td>44.8<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>43.1<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>38.8<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>39<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>38<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>32.7<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>30.1<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>29.3<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>27<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>25.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1621529\">\n<div id=\"fs-id1713613\">\n<table id=\"Table_04_03_18\" class=\"unnumbered\" summary=\"This table has two rows and seven columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 21, 25, 30, 31, 40, 50. The values in the second row are: 17, 11, 2, -1, -18, -40.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>21<\/td>\n<td>25<\/td>\n<td>30<\/td>\n<td>31<\/td>\n<td>40<\/td>\n<td>50<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>17<\/td>\n<td>11<\/td>\n<td>2<\/td>\n<td>\u20131<\/td>\n<td>\u201318<\/td>\n<td>\u201340<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1704740\">[latex]y=-\\text{1}.\\text{981}x+\\text{6}0.\\text{197;}[\/latex][latex]r=-0.\\text{998}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1301552\">\n<div id=\"fs-id2061731\">\n<table id=\"Table_04_03_19\" class=\"unnumbered\" summary=\"This table has two columns and seven rows. The first column is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first column are: 100, 80, 60, 55, 40, 20. The values in the second column are: 2000, 1798, 1589, 1580, 1390, 1202.\">\n<thead>\n<tr>\n<th><strong>[latex]x[\/latex]<\/strong><\/th>\n<th><strong>[latex]y[\/latex]<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>100<\/td>\n<td>2000<\/td>\n<\/tr>\n<tr>\n<td>80<\/td>\n<td>1798<\/td>\n<\/tr>\n<tr>\n<td>60<\/td>\n<td>1589<\/td>\n<\/tr>\n<tr>\n<td>55<\/td>\n<td>1580<\/td>\n<\/tr>\n<tr>\n<td>40<\/td>\n<td>1390<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>1202<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1549953\">\n<div id=\"fs-id1450966\">\n<table id=\"Table_04_03_20\" class=\"unnumbered\" summary=\"This table has two rows and seven columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 900, 988, 1000, 1010, 1200, 1205. The values in the second row are: 70, 80, 82, 84, 105, 108.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>900<\/td>\n<td>988<\/td>\n<td>1000<\/td>\n<td>1010<\/td>\n<td>1200<\/td>\n<td>1205<\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex]<\/td>\n<td>70<\/td>\n<td>80<\/td>\n<td>82<\/td>\n<td>84<\/td>\n<td>105<\/td>\n<td>108<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1475836\">[latex]y=0.\\text{121}x-38.841,r=0.998[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2040607\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id2066641\">\n<div id=\"fs-id1701380\">\n<p id=\"fs-id1701381\">Graph[latex]\\,f\\left(x\\right)=0.5x+10.\\,[\/latex]Pick a set of five ordered pairs using inputs[latex]\\,x=-2,\\text{1},\\text{5},\\text{6},\\text{9}\\,[\/latex]and use linear regression to verify that the function is a good fit for the data.<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1334128\">\n<p id=\"fs-id1678235\">Graph[latex]\\,f\\left(x\\right)=-2x-10.\\,[\/latex]Pick a set of five ordered pairs using inputs[latex]\\,x=-2,\\text{1},\\text{5},\\text{6},\\text{9}\\,[\/latex]and use linear regression to verify the function.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1452340\">[latex]\\left(-2,-6\\right),\\left(1,\\text{\u221212}\\right),\\left(5,-20\\right),\\left(6,\\text{\u221222}\\right),\\left(9,\\text{\u221228}\\right);\\,[\/latex]Yes, the function is a good fit.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2476616\">For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years:<\/p>\n<div id=\"fs-id2084651\" class=\"unnumbered aligncenter\">[latex]\\left(\\text{46},600\\right),\\left(\\text{48},\\text{55}0\\right),\\left(50,505\\right),\\left(\\text{52},\\text{54}0\\right),\\left(\\text{54},\\text{495}\\right).[\/latex]<\/div>\n<div id=\"fs-id1288856\">\n<div id=\"fs-id1288858\">\n<p id=\"fs-id2097965\">Use linear regression to determine a function[latex]\\,P\\,[\/latex]where the profit in thousands of dollars depends on the number of units sold in hundreds.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2178129\">\n<div id=\"fs-id2178130\">\n<p id=\"fs-id2406756\">Find to the nearest tenth and interpret the <em>x<\/em>-intercept.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2490923\">[latex]\\left(\\text{189}.8,0\\right)\\,[\/latex]If 18,980 units are sold, the company will have a profit of zero dollars.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1841800\">\n<div id=\"fs-id1579613\">\n<p id=\"fs-id1579614\">Find to the nearest tenth and interpret the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2429851\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id2673753\">For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years:<\/p>\n<p id=\"fs-id2459368\">[latex]\\left(\\text{25}00,2000\\right),\\left(\\text{265}0,2001\\right),\\left(3000,2003\\right),\\left(\\text{35}00,2006\\right),\\left(\\text{42}00,2010\\right)[\/latex]<\/p>\n<div id=\"fs-id2651915\">\n<div id=\"fs-id2111924\">\n<p id=\"fs-id2111925\">Use linear regression to determine a function[latex]\\,y,[\/latex]where the year depends on the population. Round to three decimal places of accuracy.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1794393\">[latex]y=0.00587x+\\text{1985}.4\\text{1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2621050\">\n<div id=\"fs-id2621051\">\n<p id=\"fs-id1469913\">Predict when the population will hit 8,000.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1901427\">For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:<\/p>\n<p id=\"fs-id1798995\">[latex]\\left(\\text{46},\\text{25}0\\right),\\left(\\text{48},\\text{3}05\\right),\\left(50,\\text{35}0\\right),\\left(\\text{52},\\text{39}0\\right),\\left(\\text{54},\\text{41}0\\right).[\/latex]<\/p>\n<div id=\"fs-id2202291\">\n<div id=\"fs-id2634058\">\n<p id=\"fs-id2634059\">Use linear regression to determine a function <em>y<\/em>, where the profit in thousands of dollars depends on the number of units sold in hundreds.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2523562\">[latex]y=\\text{2}0.\\text{25}x-\\text{671}.\\text{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2095664\">\n<div id=\"fs-id2095665\">\n<p id=\"fs-id2522917\">Predict when the profit will exceed one million dollars.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2227780\">For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years:<\/p>\n<p id=\"fs-id2227782\">[latex]\\left(\\text{46},\\text{25}0\\right),\\left(\\text{48},\\text{225}\\right),\\left(50,\\text{2}05\\right),\\left(\\text{52},\\text{18}0\\right),\\left(\\text{54},\\text{165}\\right).[\/latex]<\/p>\n<div id=\"fs-id2575116\">\n<div id=\"fs-id1592705\">\n<p id=\"fs-id1592706\">Use linear regression to determine a function <em>y<\/em>, where the profit in thousands of dollars depends on the number of units sold in hundreds.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2591902\">[latex]y=-\\text{1}0.\\text{75}x+\\text{742}.\\text{5}0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2395926\">\n<div id=\"fs-id2395927\">\n<p id=\"fs-id2395928\">Predict when the profit will dip below the $25,000 threshold.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3638138\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id2521178\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f48a9644-4329-4387-9b38-4ac039f12570\">Linear Functions<\/a><\/h4>\n<div id=\"fs-id2806383\">\n<div id=\"fs-id2806384\">\n<p id=\"fs-id2477392\">Determine whether the algebraic equation is linear.[latex]\\,2x+3y=7[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2095696\">Yes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2779858\">\n<div id=\"fs-id2779859\">\n<p id=\"fs-id2779860\">Determine whether the algebraic equation is linear.[latex]\\,6{x}^{2}-y=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2372659\">\n<div id=\"fs-id2372660\">\n<p id=\"fs-id2372662\">Determine whether the function is increasing or decreasing.<\/p>\n<p id=\"fs-id2060828\">[latex]f\\left(x\\right)=7x-2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1625952\">Increasing<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1625955\">\n<div id=\"fs-id2755050\">\n<p id=\"fs-id2755051\">Determine whether the function is increasing or decreasing.<\/p>\n<p id=\"fs-id2490966\">[latex]g\\left(x\\right)=-x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2275942\">\n<div id=\"fs-id2275943\">\n<p id=\"fs-id2275944\">Given each set of information, find a linear equation that satisfies the given conditions, if possible.<\/p>\n<p id=\"fs-id2560758\">Passes through[latex]\\,\\left(\\text{7},\\text{5}\\right)\\,[\/latex]and[latex]\\,\\left(\\text{3},\\text{17}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2589261\">[latex]y=-\\text{3}x+\\text{26}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2523185\">\n<div id=\"fs-id2241203\">\n<p id=\"fs-id2241204\">Given each set of information, find a linear equation that satisfies the given conditions, if possible.<\/p>\n<p id=\"fs-id1726321\"><em>x<\/em>-intercept at[latex]\\,\\left(\\text{6},0\\right)\\,[\/latex]and <em>y<\/em>-intercept at[latex]\\,\\left(0,\\text{1}0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2748158\">\n<div id=\"fs-id2748160\">\n<p id=\"fs-id2748161\">Find the slope of the line shown in the graph.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/603f3fb474ccfdcae6d7e540b637877ec40c0507\/CNX_Precalc_Figure_02_04_211.jpg\" alt=\"This is a graph of an increasing line with a y-intercept of -3 and x-intercept of 1 on an x, y coordinate plane. The x and y-axis range from -6 to 6.\" \/><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2555716\">3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2555719\">\n<div id=\"fs-id2571097\">\n<p id=\"fs-id2571098\">Find the slope of the line graphed.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/3b7ee2fb2ebdee991c1f842fdb4e7afcbbeead9f\/CNX_Precalc_Figure_02_04_212.jpg\" alt=\"This is a graph of a line with a y-intercept of -2 and no x-intercepts on an x, y coordinate plane. The x- and y-axis range from -6 to 6\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2799886\">\n<div id=\"fs-id2799888\">\n<p id=\"fs-id1433211\">Write an equation in slope-intercept form for the line shown.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/5863d8fbf42fecda8b272b7bc10a9e3533ddd82d\/CNX_Precalc_Figure_02_04_213.jpg\" alt=\"This is a graph of a line with a y-intercept of -2 and x-intercept of 1 on an x, y coordinate plane. The x- and y-axis both range from -6 to 6.\" \/><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2134830\">[latex]y=\\text{2}x-\\text{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2568491\">\n<div id=\"fs-id2568492\">\n<p id=\"fs-id1828687\">Does the following table represent a linear function? If so, find the linear equation that models the data.<\/p>\n<table id=\"Table_04_03_21\" class=\"unnumbered\" summary=\"This table has two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cg(x)\u201d. The values in the first row are: -4, 0, 2, 10. The values in the second row are: 18, -2, -12, -52.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>\u20134<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td><strong><em>g(x)<\/em><\/strong><\/td>\n<td>18<\/td>\n<td>\u20132<\/td>\n<td>\u201312<\/td>\n<td>\u201352<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2193041\">\n<div id=\"fs-id2193042\">\n<p id=\"fs-id2193043\">Does the following table represent a linear function? If so, find the linear equation that models the data.<\/p>\n<table id=\"Table_04_03_22\" class=\"unnumbered\" summary=\"This table has two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201c\u201d and the second is labeled: \u201cg(x)\u201d. The values in the first row are: 6, 8, 12, 26. The values in the second row are: -8, -12, -18, -46.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>12<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong><em>g(x)<\/em><\/strong><\/td>\n<td>\u20138<\/td>\n<td>\u201312<\/td>\n<td>\u201318<\/td>\n<td>\u201346<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2113224\">Not linear.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2241122\">\n<div id=\"fs-id2241123\">\n<p id=\"fs-id2241124\">On June 1<sup>st<\/sup>, a company has $4,000,000 profit. If the company then loses 150,000 dollars per day thereafter in the month of June, what is the company\u2019s profit <em>n<sup>th<\/sup><\/em>day after June 1<sup>st<\/sup>?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2078767\">For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:<\/p>\n<div id=\"fs-id2468288\">\n<div id=\"fs-id2468289\">\n<p id=\"fs-id2468290\">[latex]\\begin{array}{c}2x-6y=12\\\\ -x+3y=1\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2757779\">parallel<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2757782\">\n<div id=\"fs-id2637956\">\n<p id=\"fs-id2637957\">[latex]\\begin{array}{c}y=\\frac{1}{3}x-2\\\\ 3x+y=-9\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2616282\">For the following exercises, find the <em>x<\/em>&#8211; and <em>y<\/em>&#8211; intercepts of the given equation<\/p>\n<div id=\"fs-id2489259\">\n<div id=\"fs-id2489260\">\n<p id=\"fs-id2489262\">[latex]7x+9y=-63[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2459233\">[latex]\\left(\u20139,0\\right);\\left(0,\u20137\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2779958\">\n<div id=\"fs-id2779959\">\n<p id=\"fs-id1600522\">[latex]f\\left(x\\right)=2x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2128502\">For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?<\/p>\n<div id=\"fs-id1828556\">\n<div id=\"fs-id1828557\">\n<p id=\"fs-id1828558\">Line 1: Passes through[latex]\\,\\left(5,11\\right)\\,[\/latex]and[latex]\\,\\left(10,1\\right)[\/latex]<\/p>\n<p id=\"fs-id1623644\">Line 2: Passes through[latex]\\,\\left(-1,3\\right)\\,[\/latex]and[latex]\\,\\left(-5,11\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1718504\">Line 1:[latex]\\,m=-2;[\/latex]Line 2:[latex]\\,m=-2;[\/latex]Parallel<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id889716\">\n<div id=\"fs-id889717\">\n<p id=\"fs-id889718\">Line 1: Passes through[latex]\\,\\left(8,-10\\right)\\,[\/latex]and[latex]\\,\\left(0,-26\\right)[\/latex]<\/p>\n<p id=\"fs-id1598366\">Line 2: Passes through[latex]\\,\\left(2,5\\right)\\,[\/latex]and[latex]\\,\\left(4,4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2420694\">\n<div id=\"fs-id2480225\">\n<p id=\"fs-id2480226\">Write an equation for a line perpendicular to[latex]\\,f\\left(x\\right)=5x-1\\,[\/latex]and passing through the point (5, 20).<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2202404\">[latex]y=-0.2x+21[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1675129\">\n<div id=\"fs-id2779876\">\n<p id=\"fs-id2779877\">Find the equation of a line with a <em>y<\/em>&#8211; intercept of[latex]\\,\\left(0,2\\right)\\,[\/latex]and slope[latex]\\,-\\frac{1}{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1694239\">\n<div id=\"fs-id1694240\">\n<p id=\"fs-id1694242\">Sketch a graph of the linear function[latex]\\,f\\left(t\\right)=2t-5.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/09f3d933b61b5a18631e559c4be70c86977de017\/CNX_Precalc_Figure_02_04_214.jpg\" alt=\"This is a graph of f of t = 2 times t minus 5 on a x, y coordinate plane. The x-axis ranges from -4 to 6 and the y-axis ranges from -6 to 6. The curve is an increasing linear function that goes through the points (0,-5) and (2.5,0).\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2489570\">\n<div id=\"fs-id2489571\">\n<p id=\"fs-id1966917\">Find the point of intersection for the 2 linear functions:[latex]\\,\\begin{array}{c}x=y+6\\\\ 2x-y=13\\end{array}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2062262\">\n<div id=\"fs-id2062263\">\n<p id=\"fs-id2395879\">A car rental company offers two plans for renting a car.<\/p>\n<p id=\"fs-id2395882\">Plan A: 25 dollars per day and 10 cents per mile<\/p>\n<p id=\"fs-id2486072\">Plan B: 50 dollars per day with free unlimited mileage<\/p>\n<p>How many miles would you need to drive for plan B to save you money?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1093901\">More than 250<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1093904\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/dcf784dc-acd1-46a8-b5b0-cd5d8673b83a\">Modeling with Linear Functions<\/a><\/h4>\n<div id=\"fs-id1293909\">\n<div id=\"fs-id1293910\">\n<p id=\"fs-id1293911\">Find the area of a triangle bounded by the <em>y<\/em> axis, the line[latex]\\,f\\left(x\\right)=10-2x,[\/latex]and the line perpendicular to[latex]\\,f\\,[\/latex]that passes through the origin.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2063497\">\n<div id=\"fs-id2063498\">\n<p id=\"fs-id2063499\">A town\u2019s population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2276060\">118,000<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1843378\">\n<div id=\"fs-id1843379\">\n<p id=\"fs-id1843380\">The number of people afflicted with the common cold in the winter months dropped steadily by 50 each year since 2004 until 2010. In 2004, 875 people were inflicted.<\/p>\n<p id=\"fs-id1484577\">Find the linear function that models the number of people afflicted with the common cold <em>C<\/em> as a function of the year,[latex]\\,t.\\,[\/latex]When will no one be afflicted?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2748176\">For the following exercises, use the graph in <a class=\"autogenerated-content\" href=\"#Figure_04_03_202\">(Figure)<\/a> showing the profit,[latex]\\,y,[\/latex]in thousands of dollars, of a company in a given year,[latex]\\,x,[\/latex]where[latex]\\,x\\,[\/latex]represents years since 1980.<\/p>\n<figure style=\"width: 291px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/67e128c421b4c05cd0a4b2e5c6acdf9b3b796f28\/CNX_Precalc_Figure_02_04_215.jpg\" alt=\"This graph shows profits starting at 1985 at $10,000 and ending at 2005 at $4,000. The x-axis ranges from 0 to 30 in intervals of 5 and the y \u2013axis goes from 0 to 12,000 in intervals of 2,000.\" width=\"291\" height=\"254\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/figcaption><\/figure>\n<div id=\"fs-id1429436\">\n<div id=\"fs-id1429437\">\n<p id=\"fs-id2662623\">Find the linear function <em>y<\/em>, where <em>y<\/em> depends on[latex]\\,x,[\/latex]the number of years since 1980.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2637049\">[latex]y=-\\text{3}00x+\\text{11},\\text{5}00[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2527720\">\n<div id=\"fs-id2527721\">\n<p id=\"fs-id2527722\">Find and interpret the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2552031\">For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.<\/p>\n<div id=\"fs-id2518478\">\n<div id=\"fs-id2518479\">\n<p id=\"fs-id2518480\">Assume the population is changing linearly.<\/p>\n<ol id=\"fs-id2518483\" type=\"a\">\n<li>How much did the population grow between the year 2004 and 2012?<\/li>\n<li>What is the average population growth per year?<\/li>\n<li>Find an equation for the population, <em>P<\/em>, of the school <em>t<\/em> years after 2004.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2468246\">a) 800 b) 100 students per year c)[latex]\\,P\\left(t\\right)=\\text{1}00t+\\text{17}00[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id888455\">For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.<\/p>\n<div id=\"fs-id2424214\">\n<div id=\"fs-id2424215\">\n<p id=\"fs-id888457\">Find a formula for the moose population,[latex]\\,P.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2411048\">\n<div id=\"fs-id1016508\">\n<p id=\"fs-id1016509\">What does your model predict the moose population to be in 2020?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1016513\">18,500<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2637890\">For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in <a class=\"autogenerated-content\" href=\"#Table_04_03_23\">(Figure)<\/a>. Assume that the house values are changing linearly.<\/p>\n<table id=\"Table_04_03_23\" summary=\"This table has three rows and three columns. The first column is labeled: \u201cYear\u201d the second column is labeled: \u201cPima Central\u201d and the third column is labeled: \u201cEast Valley\u201d. The values for the first column are: 1970 and 2010. The values for the second column are: 32,000 and 85,000. The values for the third column are: 120,250 and 150,000.\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Pima Central<\/th>\n<th>East Valley<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1970<\/td>\n<td>32,000<\/td>\n<td>120,250<\/td>\n<\/tr>\n<tr>\n<td>2010<\/td>\n<td>85,000<\/td>\n<td>150,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1802777\">\n<div id=\"fs-id1802778\">\n<p id=\"fs-id1802779\">In which subdivision have home values increased at a higher rate?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1855668\">\n<div id=\"fs-id1855669\">\n<p id=\"fs-id1855670\">If these trends were to continue, what would be the median home value in Pima Central in 2015?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1794327\">$91,625<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2267333\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/e9d5f844-6760-4f85-8f79-67292a6703dc\">Fitting Linear Models to Data<\/a><\/h4>\n<div id=\"fs-id2395889\">\n<div id=\"fs-id2395890\">\n<p id=\"fs-id2395891\">Draw a scatter plot for the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_24\">(Figure)<\/a>. Then determine whether the data appears to be linearly related.<\/p>\n<table id=\"Table_04_03_24\" summary=\"This table shows two rows and six columns. The values in the first row are: 0, 2, 4, 6, 8, 10. The values in the second row are: -105, -50, 1, 55, 105, 160.\">\n<tbody>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u2013105<\/td>\n<td>\u201350<\/td>\n<td>1<\/td>\n<td>55<\/td>\n<td>105<\/td>\n<td>160<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id1739382\">\n<div id=\"fs-id1739383\">\n<p id=\"fs-id1739384\">Draw a scatter plot for the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_25\">(Figure)<\/a>. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation?<\/p>\n<table id=\"Table_04_03_25\" summary=\"This table has two columns and six rows. The first column is labeled: \u201cYear\u201d and the second is labeled: \u201cPopulation\u201d. The values in the first column are: 1990, 1995, 2000, 2005, 2010. The values in the second column are: 5,600; 5,950; 6,300; 6,600; 6,900.\">\n<thead>\n<tr>\n<th><strong>Year<\/strong><\/th>\n<th><strong>Population<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1990<\/td>\n<td>5,600<\/td>\n<\/tr>\n<tr>\n<td>1995<\/td>\n<td>5,950<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>6,300<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>6,600<\/td>\n<\/tr>\n<tr>\n<td>2010<\/td>\n<td>6,900<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2208906\">Extrapolation<img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/8beebd5bb1c6ca6c8de43339354cf3293128697c\/CNX_Precalc_Figure_02_04_226.jpg\" alt=\"Scatter plot with the points (1990,5600); (1995,5950); (2000,6300); (2005,6600); and (2010,6900).\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2267039\">\n<div id=\"fs-id2267040\">\n<p id=\"fs-id2267041\">Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in <a class=\"autogenerated-content\" href=\"#Table_04_03_26\">(Figure)<\/a>. Plot the points, then sketch a line that fits the data.<\/p>\n<table id=\"Table_04_03_26\" summary=\"This table shows two columns and nine rows. The first column is labeled: \u201cPredicted\u201d and the second is labeled: \u201cActual\u201d. The values in the first column are: 6, 7, 7, 8, 7, 9, 10, 10. The values in the second column are: 6, 7, 8, 8, 9, 10, 10, 9.\">\n<thead>\n<tr>\n<th><strong>Predicted<\/strong><\/th>\n<th><strong>Actual<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>6<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2578689\">\n<div id=\"fs-id2578690\">\n<p id=\"fs-id2578691\">Draw a best-fit line for the plotted data.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/efcc88d43851ec9b3dd4b1fa62e1491fc2a314ad\/CNX_Precalc_Figure_02_04_216.jpg\" alt=\"Scatter plot of the points: (2,78); (4,81); (6,85); (8,90); and (10,99).\" \/><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/7b9b0a87d04e3bb08bc86beddd2023daa447fb01\/CNX_Precalc_Figure_02_04_217.jpg\" alt=\"Scatter plot of: (2,78); (4,81); (6,85); (8,90); and (10,99) and the line of best fit running through these points. The line of best fit goes through most of the points.\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2755468\">For the following exercises, consider the data in <a class=\"autogenerated-content\" href=\"#Table_04_03_27\">(Figure)<\/a>, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year.<\/p>\n<table id=\"Table_04_03_27\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cYear\u201d and the second is labeled: \u201cPercent Graduates\u201d. The values in the first row are: 2000, 2002, 2005, 2007, 2010. The values in the second row are: 6.5, 7.0, 7.4, 8.2, 9.0\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2000<\/td>\n<td>2002<\/td>\n<td>2005<\/td>\n<td>2007<\/td>\n<td>2010<\/td>\n<\/tr>\n<tr>\n<td><strong>Percent Graduates<\/strong><\/td>\n<td>6.5<\/td>\n<td>7.0<\/td>\n<td>7.4<\/td>\n<td>8.2<\/td>\n<td>9.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id2489600\">\n<div id=\"fs-id2489245\">\n<p id=\"fs-id2489246\">Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2552236\">\n<div id=\"fs-id2552237\">\n<p id=\"fs-id2552238\">In what year will the percentage exceed 12%?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2448110\">2023<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1351720\">Based on the set of data given in <a class=\"autogenerated-content\" href=\"#Table_04_03_28\">(Figure)<\/a>, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.<\/p>\n<table id=\"Table_04_03_28\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values of the first row are: 17, 20, 23, 26, 29. The values of the second row are: 15, 25, 31, 37, 40.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>17<\/td>\n<td>20<\/td>\n<td>23<\/td>\n<td>26<\/td>\n<td>29<\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex]<\/td>\n<td>15<\/td>\n<td>25<\/td>\n<td>31<\/td>\n<td>37<\/td>\n<td>40<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2674532\">\n<div id=\"fs-id2164258\">\n<p id=\"fs-id2164259\">Based on the set of data given in <a class=\"autogenerated-content\" href=\"#Table_04_03_29\">(Figure)<\/a>, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.<\/p>\n<table id=\"Table_04_03_29\" summary=\"This table has two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values of the first row are: 10, 12, 15, 18, 20. The values of the second row are: 36, 34, 30, 28, 22.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>10<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<td>18<\/td>\n<td>20<\/td>\n<\/tr>\n<tr>\n<td>[latex]y[\/latex]<\/td>\n<td>36<\/td>\n<td>34<\/td>\n<td>30<\/td>\n<td>28<\/td>\n<td>22<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2489280\">[latex]y=-1.294x+49.412; r=-0.974[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2589276\">For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years:<\/p>\n<p id=\"fs-id1802875\">[latex]\\left(\\text{3,6}00,2000\\right);\\left(\\text{4,}000,2001\\right);\\left(\\text{4,7}00,2003\\right);\\left(\\text{6,}000,2006\\right)[\/latex]<\/p>\n<div id=\"fs-id1967151\">\n<div id=\"fs-id1967152\">\n<p id=\"fs-id1967153\">Use linear regression to determine a function[latex]\\,y,[\/latex]where the year depends on the population, to three decimal places of accuracy.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2574347\">\n<div id=\"fs-id2574348\">\n<p id=\"fs-id2574350\">Predict when the population will hit 12,000.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2275847\">2027<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2275850\">\n<div id=\"fs-id2275852\">\n<p id=\"fs-id2275853\">What is the correlation coefficient for this model to three decimal places of accuracy?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1505691\">\n<div id=\"fs-id1505692\">\n<p id=\"fs-id1505693\">According to the model, what is the population in 2014?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1505698\">7,660<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1505702\" class=\"practice-test\">\n<h3>Chapter Practice Test<\/h3>\n<div id=\"fs-id2560601\">\n<div id=\"fs-id2560602\">\n<p id=\"fs-id2560603\">Determine whether the following algebraic equation can be written as a linear function.[latex]\\,2x+3y=7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1803014\">\n<div id=\"fs-id1803015\">\n<p id=\"fs-id1803016\">Determine whether the following function is increasing or decreasing.[latex]\\,f\\left(x\\right)=-2x+5[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2176627\">Decreasing<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2560692\">\n<div id=\"fs-id2560693\">\n<p id=\"fs-id2560694\">Determine whether the following function is increasing or decreasing.[latex]f\\left(x\\right)=7x+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2424337\">\n<div id=\"fs-id2424338\">\n<p id=\"fs-id2489227\">Find a linear equation that passes through (5, 1) and (3, \u20139), if possible.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2489233\">[latex]y=5x-24[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1967119\">\n<div id=\"fs-id1967120\">\n<p id=\"fs-id2429855\">Find a linear equation, that has an <em>x<\/em> intercept at (\u20134, 0) and a <em>y<\/em>-intercept at (0, \u20136), if possible.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2489441\">\n<div id=\"fs-id2489442\">\n<p id=\"fs-id2757895\">Find the slope of the line in <a class=\"autogenerated-content\" href=\"#Image_04_03_220\">(Figure)<\/a>.<\/p>\n<div id=\"Image_04_03_220\" class=\"small\">\n<figure style=\"width: 438px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/250ab34d33c71d7624004041bea502b7f941ee49\/CNX_Precalc_Figure_02_04_218.jpg\" alt=\"This image is a graph of a decreasing linear function on an x, y coordinate plane. The x and y-axis range from -6 to 6. The line passes through the points (0,2) and (1,0).\" width=\"438\" height=\"437\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2136527\">[latex]m=-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2429836\">\n<div id=\"fs-id2429838\">\n<p id=\"fs-id2429839\">Write an equation for line in <a class=\"autogenerated-content\" href=\"#Image_04_03_221\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 438px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/415a3220bb8b54ff57fb4e6e4e0339e054952dc1\/CNX_Precalc_Figure_02_04_219.jpg\" alt=\"This image is a graph showing a decreasing linear function on an x, y coordinate plane. The x and y axis range from -6 to 6. The line passes through the points (0,-1) and (-.5,0).\" width=\"438\" height=\"437\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<div id=\"Image_04_03_221\" class=\"small\">Does <a class=\"autogenerated-content\" href=\"#Table_04_03_30\">(Figure)<\/a> represent a linear function? If so, find a linear equation that models the data.<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1795368\">\n<div id=\"fs-id1795369\">\n<table id=\"Table_04_03_30\" summary=\"The table shows two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cg(x)\u201d. The values of the first row are: -6, 0, 2, 4. The values of the second row are: 14, 32, 38, 44.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>\u20136<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>14<\/td>\n<td>32<\/td>\n<td>38<\/td>\n<td>44<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2590722\">Yes,[latex]\\,y=3x+32[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2448065\">\n<div id=\"fs-id2674652\">\n<p id=\"fs-id2448066\">Does <a class=\"autogenerated-content\" href=\"#Table_04_03_31\">(Figure)<\/a> represent a linear function? If so, find a linear equation that models the data.<\/p>\n<table id=\"Table_04_03_31\" summary=\"The table shows two rows and five columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cg(x)\u201d. The values of the first row are: 1, 3, 7, 11. The values of the second row are: 4, 9, 19, 12.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td><strong><em>g<\/em>(<em>x<\/em>)<\/strong><\/td>\n<td>4<\/td>\n<td>9<\/td>\n<td>19<\/td>\n<td>12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div id=\"fs-id2448041\">\n<div id=\"fs-id2448042\">\n<p id=\"fs-id2448044\">At 6 am, an online company has sold 120 items that day. If the company sells an average of 30 items per hour for the remainder of the day, write an expression to represent the number of items that were sold[latex]\\,n\\,[\/latex]after 6 am.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2549693\">[latex]y=30x+120[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2241342\">For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.<\/p>\n<div id=\"fs-id2241346\">\n<div id=\"fs-id2241347\">\n<p id=\"fs-id2241348\">[latex]\\begin{array}{c}y=\\frac{3}{4}x-9\\\\ -4x-3y=8\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2674691\">\n<div id=\"fs-id2674692\">\n<p id=\"fs-id2674693\">[latex]\\begin{array}{c}-2x+y=3\\\\ 3x+\\frac{3}{2}y=5\\end{array}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2590764\">Neither<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2590768\">\n<div id=\"fs-id2590769\">\n<p id=\"fs-id2590770\">Find the <em>x<\/em>&#8211; and <em>y<\/em>-intercepts of the equation[latex]\\,2x+7y=-14.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2489142\">\n<div id=\"fs-id2489143\">\n<p id=\"fs-id2489144\">Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is the pair of lines parallel, perpendicular, or neither?<\/p>\n<p id=\"fs-id2489148\">Line 1: Passes through[latex]\\,\\left(-2,-6\\right)\\,[\/latex]and[latex]\\,\\left(3,14\\right)[\/latex]<\/p>\n<p id=\"fs-id1505527\">Line 2: Passes through[latex]\\,\\left(2,6\\right)\\,[\/latex]and[latex]\\,\\left(4,14\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2365321\">Line 1:[latex]\\,m=4;[\/latex]Line 2:[latex]\\,m=4;[\/latex]parallel<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2267122\">\n<div id=\"fs-id2267123\">\n<p id=\"fs-id2267124\">Write an equation for a line perpendicular to[latex]\\,f\\left(x\\right)=4x+3\\,[\/latex]and passing through the point[latex]\\,\\left(8,10\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2437806\">\n<div id=\"fs-id2437807\">\n<p id=\"fs-id1720532\">Sketch a line with a <em>y<\/em>-intercept of[latex]\\,\\left(0,\\text{5}\\right)\\,[\/latex]and slope[latex]\\,-\\frac{5}{2}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/db97384e013f70779f231db0874722ef0d27ddf7\/CNX_Precalc_Figure_02_04_220.jpg\" alt=\"This image is a graph showing a decreasing linear function on an x, y coordinate plane. The x and y-axis range from -6 to 6. The line passes through the points (0,5) and (2,0) and a slope of: -5\/2.\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1801268\">\n<div id=\"fs-id1801269\">\n<p id=\"fs-id2406692\">Graph of the linear function[latex]\\,f\\left(x\\right)=-x+6.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2800116\">\n<div id=\"fs-id2800118\">\n<p id=\"fs-id2800119\">For the two linear functions, find the point of intersection:[latex]\\begin{array}{c}x=y+2\\\\ 2x-3y=-1\\end{array}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2436158\">[latex]\\left(7,5\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2436183\">\n<div id=\"fs-id2436184\">\n<p id=\"fs-id2436185\">A car rental company offers two plans for renting a car.<\/p>\n<p id=\"fs-id2436188\">Plan A: $25 per day and $0.10 per mile<\/p>\n<p id=\"fs-id2436192\">Plan B: $40 per day with free unlimited mileage<\/p>\n<p id=\"fs-id2436195\">How many miles would you need to drive for plan B to save you money?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2436202\">\n<div id=\"fs-id2436203\">\n<p id=\"fs-id2436204\">Find the area of a triangle bounded by the <em>y<\/em> axis, the line[latex]\\,f\\left(x\\right)=12-4x,[\/latex]and the line perpendicular to[latex]\\,f\\,[\/latex]that passes through the origin.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2804285\">16.94 square units<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2804288\">\n<div id=\"fs-id2804289\">\n<p id=\"fs-id2804290\">A town\u2019s population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2804301\">\n<div id=\"fs-id2804302\">\n<p id=\"fs-id2804303\">The number of people afflicted with the common cold in the winter months dropped steadily by 25 each year since 2002 until 2012. In 2002, 8,040 people were inflicted. Find the linear function that models the number of people afflicted with the common cold[latex]\\,C\\,[\/latex]as a function of the year,[latex]\\,t.\\,[\/latex]When will less than 6,000 people be afflicted?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2436241\">2083<\/p>\n<\/details>\n<\/div>\n<\/div>\n<figure style=\"width: 291px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/bb29c461e3cdf022bfd0e468c15d2745a5280fb8\/CNX_Precalc_Figure_02_04_222.jpg\" alt=\"This image is a graph showing the company's profit from 1985 at around $15,000 to 2010 at about $32,500. The x-axis goes from 0 to 30 in intervals of 5 and the y-axis goes from 0 to 35,000 in intervals of 5,000.\" width=\"291\" height=\"285\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id2436244\">For the following exercises, use the graph in <a class=\"autogenerated-content\" href=\"#Figure_04_03_224\">(Figure)<\/a>, showing the profit,[latex]y,[\/latex]in thousands of dollars, of a company in a given year,[latex]\\,x,[\/latex]where[latex]\\,x\\,[\/latex]represents years since 1980.<\/p>\n<div id=\"fs-id2436305\">\n<div id=\"fs-id2436306\">\n<p id=\"fs-id2436307\">Find the linear function[latex]\\,y,[\/latex]where[latex]\\,y\\,[\/latex]depends on[latex]\\,x,[\/latex]the number of years since 1980.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2363849\">\n<div id=\"fs-id2363850\">\n<p id=\"fs-id2363851\">Find and interpret the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2363860\">[latex]\\left(0,10,625\\right);[\/latex]In 1980, the profit was $10,625,000.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2387148\">\n<div id=\"fs-id2387149\">\n<p id=\"fs-id2387150\">In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly.<\/p>\n<ol id=\"fs-id2387155\" type=\"a\">\n<li>How much did the population drop between the year 2004 and 2012?<\/li>\n<li>What is the average population decline per year?<\/li>\n<li>Find an equation for the population, <em>P<\/em>, of the school <em>t<\/em> years after 2004.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id2387205\">\n<div id=\"fs-id2387206\">\n<p id=\"fs-id2387207\">Draw a scatter plot for the data provided in <a class=\"autogenerated-content\" href=\"#Table_04_03_32\">(Figure)<\/a>. Then determine whether the data appears to be linearly related.<\/p>\n<table id=\"Table_04_03_32\" summary=\"This table shows two rows and six columns. The values in the first row are: 0, 2, 4, 6, 8, 10. The values in the second row are: -450, -200, 10, 265, 500 and 755.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>\u2013450<\/td>\n<td>\u2013200<\/td>\n<td>10<\/td>\n<td>265<\/td>\n<td>500<\/td>\n<td>755<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/c7a24d60ff818a189bac3a4fdd21e60511778fc7\/CNX_Precalc_Figure_02_04_231.jpg\" alt=\"Scatterplot with a collection of points: (0,-450); (2,-200); (4,10); (6,265); (8,500) and (10,755). The data appears linear.\" \/><\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id2804427\">\n<div id=\"fs-id2804428\">\n<p id=\"fs-id2804429\">Draw a best-fit line for the plotted data.<img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/9cf59d69cd7c32d0cd7612f75bcd0f2ed9f0b685\/CNX_Precalc_Figure_02_04_223.jpg\" alt=\"Scatterplot with domain from 2 to 10 and range from 20 from 33. The points plotted are (2,20); (4,23); (6,26); (8,26); and (10,32).\" \/><\/p>\n<p>For the following exercises, use <a class=\"autogenerated-content\" href=\"#Table_04_03_33\">(Figure)<\/a>, which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year.<\/p>\n<\/div>\n<\/div>\n<table id=\"Table_04_03_33\" summary=\"This table shows two rows and six columns. The first row is labeled: \u201cYear\u201d and the second is labeled: \u201cPercent Graduates\u201d. The values in the first row are: 2000, 2002, 2005, 2007, 2010. The values in the second row are: 8.5, 8.0, 7.2, 6.7 and 6.4\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2000<\/td>\n<td>2002<\/td>\n<td>2005<\/td>\n<td>2007<\/td>\n<td>2010<\/td>\n<\/tr>\n<tr>\n<td><strong>Percent Graduates<\/strong><\/td>\n<td>8.5<\/td>\n<td>8.0<\/td>\n<td>7.2<\/td>\n<td>6.7<\/td>\n<td>6.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id2406470\">\n<div id=\"fs-id2406471\">\n<p id=\"fs-id2406472\">Determine whether the trend appears linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/88451207483f635c2c1c476a528e3135da0992b0\/CNX_Precalc_Figure_02_04_232.jpg\" alt=\"Scatterplot with a collection of points: (2000,8.5); (2002,8); (2005,7.2); (2007,6.7); and (2010,6.4). The data appears linear\" \/><\/p>\n<p id=\"fs-id2406495\">[latex]y=-0.219x+445.990[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2363667\">\n<div id=\"fs-id2406523\">\n<p id=\"fs-id2406524\">In what year will the percentage drop below 4%?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2406532\">\n<div id=\"fs-id2406533\">\n<p id=\"fs-id2406534\">Based on the set of data given in <a class=\"autogenerated-content\" href=\"#Table_04_03_34\">(Figure)<\/a>, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient. Round to three decimal places of accuracy.<\/p>\n<table id=\"Table_04_03_34\" summary=\"This table shows two rows and six columns. The first row is labeled: \u201cx\u201d and the second is labeled: \u201cy\u201d. The values in the first row are: 16, 18, 20, 24, 26. The values in the second row are: 106, 110, 115, 120, 125.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><strong><em>x<\/em><\/strong><\/td>\n<td>16<\/td>\n<td>18<\/td>\n<td>20<\/td>\n<td>24<\/td>\n<td>26<\/td>\n<\/tr>\n<tr>\n<td><strong><em>y<\/em><\/strong><\/td>\n<td>106<\/td>\n<td>110<\/td>\n<td>115<\/td>\n<td>120<\/td>\n<td>125<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2406661\">[latex]y=1.820x+77.349,r=0.986[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2406691\">For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:<\/p>\n<p id=\"fs-id2406697\">[latex]\\left(4,500,2000\\right);\\left(4,700,2001\\right);\\left(5,200,2003\\right);\\left(5,800,2006\\right)[\/latex]<\/p>\n<div id=\"fs-id2621231\">\n<div id=\"fs-id2621232\">\n<p id=\"fs-id2621233\">Use linear regression to determine a function <em>y<\/em>, where the year depends on the population. Round to three decimal places of accuracy.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2621264\">\n<div id=\"fs-id2621265\">\n<p id=\"fs-id2621266\">Predict when the population will hit 20,000.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2621270\">2070<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2621274\">\n<div id=\"fs-id2621275\">\n<p id=\"fs-id2621276\">What is the correlation coefficient for this model?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2621300\">\n<dt>correlation coefficient<\/dt>\n<dd id=\"fs-id2621304\">a value,[latex]\\,r,[\/latex]between \u20131 and 1 that indicates the degree of linear correlation of variables, or how closely a regression line fits a data set.<\/dd>\n<\/dl>\n<dl id=\"fs-id1672566\">\n<dt>extrapolation<\/dt>\n<dd id=\"fs-id1672569\">predicting a value outside the domain and range of the data<\/dd>\n<\/dl>\n<dl id=\"fs-id1672572\">\n<dt>interpolation<\/dt>\n<dd id=\"fs-id1672575\">predicting a value inside the domain and range of the data<\/dd>\n<\/dl>\n<dl id=\"fs-id1672579\">\n<dt>least squares regression<\/dt>\n<dd id=\"fs-id1672582\">a statistical technique for fitting a line to data in a way that minimizes the differences between the line and data values<\/dd>\n<\/dl>\n<dl id=\"fs-id1672586\">\n<dt>model breakdown<\/dt>\n<dd id=\"fs-id1672589\">when a model no longer applies after a certain point<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-74-1\">Selected data from <a href=\"http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/\">http:\/\/classic.globe.gov\/fsl\/scientistsblog\/2007\/10\/<\/a>. Retrieved Aug 3, 2010 <a href=\"#return-footnote-74-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-74-2\">Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values. <a href=\"#return-footnote-74-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><li id=\"footnote-74-3\">For example, <a href=\"http:\/\/www.shodor.org\/unchem\/math\/lls\/leastsq.html\">http:\/\/www.shodor.org\/unchem\/math\/lls\/leastsq.html<\/a> <a href=\"#return-footnote-74-3\" class=\"return-footnote\" aria-label=\"Return to footnote 3\">&crarr;<\/a><\/li><li id=\"footnote-74-4\">http:\/\/www.bts.gov\/publications\/national_transportation_statistics\/2005\/html\/table_04_10.html <a href=\"#return-footnote-74-4\" class=\"return-footnote\" aria-label=\"Return to footnote 4\">&crarr;<\/a><\/li><li id=\"footnote-74-5\">Based on data from http:\/\/www.census.gov\/hhes\/socdemo\/education\/data\/cps\/historical\/index.html. Accessed 5\/1\/2014. <a href=\"#return-footnote-74-5\" class=\"return-footnote\" aria-label=\"Return to footnote 5\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":291,"menu_order":4,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-74","chapter","type-chapter","status-publish","hentry"],"part":67,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/74","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/74\/revisions"}],"predecessor-version":[{"id":75,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/74\/revisions\/75"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/67"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/74\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=74"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=74"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=74"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=74"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}