{"id":36,"date":"2019-08-20T17:01:25","date_gmt":"2019-08-20T21:01:25","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/the-rectangular-coordinate-systems-and-graphs\/"},"modified":"2022-06-01T10:39:18","modified_gmt":"2022-06-01T14:39:18","slug":"the-rectangular-coordinate-systems-and-graphs","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/the-rectangular-coordinate-systems-and-graphs\/","title":{"raw":"The Rectangular Coordinate Systems and Graphs","rendered":"The Rectangular Coordinate Systems and Graphs"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section you will:\n<ul>\n \t<li>Plot ordered pairs in a Cartesian coordinate system.<\/li>\n \t<li>Graph equations by plotting points.<\/li>\n \t<li>Graph equations with a graphing utility.<\/li>\n \t<li>Find [latex]x[\/latex]-intercepts and [latex]y[\/latex]-intercepts.<\/li>\n \t<li>Use the distance formula.<\/li>\n \t<li>Use the midpoint formula.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_02_01_001\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132911\/CNX_CAT_Figure_02_01_001.jpg\" alt=\"Road map of a city with street names on an x, y coordinate grid. Various points are marked in red on the grid lines indicating different locations on the map.\" width=\"731\" height=\"480\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id2906377\">Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in <a class=\"autogenerated-content\" href=\"#Figure_02_01_001\">(Figure)<\/a>. Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.<\/p>\n\n<div id=\"fs-id1392675\" class=\"bc-section section\">\n<h3>Plotting Ordered Pairs in the Cartesian Coordinate System<\/h3>\n<p id=\"fs-id2500615\">An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\n<p id=\"fs-id1960277\">While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the <em>x-<\/em>axis and the vertical axis the <em>y-<\/em>axis.<\/p>\n<p id=\"fs-id1167648\">The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in <a class=\"autogenerated-content\" href=\"#Figure_02_01_002\">(Figure)<\/a><\/p>\n\n<div id=\"Figure_02_01_002\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small \"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132913\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1423341\">The center of the plane is the point at which the two axes cross. It is known as the origin, or point[latex]\\left(0,0\\right).[\/latex]From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in <a class=\"autogenerated-content\" href=\"#Figure_02_01_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_02_01_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132919\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1787344\">Each point in the plane is identified by its <em>x-<\/em>coordinate, or horizontal displacement from the origin, and its <em>y-<\/em>coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form[latex]\\,\\left(x,y\\right).\\,[\/latex]An ordered pair is also known as a coordinate pair because it consists of <em>x-<\/em> and <em>y<\/em>-coordinates. For example, we can represent the point[latex]\\,\\left(3,-1\\right)\\,[\/latex]in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_004\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_02_01_004\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132921\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id3155322\">When dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>y-<\/em>axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\n\n<div id=\"fs-id2782510\" class=\"textbox key-takeaways\">\n<h3>Cartesian Coordinate System<\/h3>\n<p id=\"fs-id1400039\">A two-dimensional plane where the<\/p>\n\n<ul id=\"fs-id573737\">\n \t<li><em>x<\/em>-axis is the horizontal axis<\/li>\n \t<li><em>y<\/em>-axis is the vertical axis<\/li>\n<\/ul>\n<p id=\"fs-id3085633\">A point in the plane is defined as an ordered pair,[latex]\\,\\left(x,y\\right),[\/latex]such that <em>x <\/em>is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.<\/p>\n\n<\/div>\n<div id=\"Example_02_01_01\" class=\"textbox examples\">\n<div id=\"fs-id2270902\">\n<div id=\"fs-id1931516\">\n<h3>Plotting Points in a Rectangular Coordinate System<\/h3>\n<p id=\"fs-id1324875\">Plot the points[latex]\\,\\left(-2,4\\right),[\/latex][latex]\\left(3,3\\right),[\/latex]and[latex]\\,\\left(0,-3\\right)\\,[\/latex]in the plane.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1718016\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1718016\"]\n<p id=\"fs-id1718016\">To plot the point[latex]\\,\\left(-2,4\\right),[\/latex]begin at the origin. The <em>x<\/em>-coordinate is \u20132, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y <\/em>direction.<\/p>\n<p id=\"fs-id2958885\">To plot the point[latex]\\,\\left(3,3\\right),[\/latex]begin again at the origin. The <em>x<\/em>-coordinate is 3, so move three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y <\/em>direction.<\/p>\n<p id=\"fs-id2433014\">To plot the point[latex]\\,\\left(0,-3\\right),[\/latex]begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction. See the graph in <a class=\"autogenerated-content\" href=\"#Figure_02_01_005\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_02_01_005\" class=\"small wp-caption aligncenter\"><span id=\"fs-id2187492\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132923\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1391320\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1215275\">Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1940663\" class=\"bc-section section\">\n<h3>Graphing Equations by Plotting Points<\/h3>\n<p id=\"fs-id1951777\">We can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n<p id=\"fs-id1811199\">Suppose we want to graph the equation[latex]\\,y=2x-1.\\,[\/latex]We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x<\/em>- and <em>y<\/em>-values is an ordered pair that can be plotted. <a class=\"autogenerated-content\" href=\"#Table_02_01_01\">(Figure)<\/a> lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.<\/p>\n\n<table id=\"Table_02_01_01\" summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=2x-1[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1386116\">We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_006\">(Figure)<\/a><strong>. <\/strong>This is not true for all equations.<\/p>\n\n<div id=\"Figure_02_01_006\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132927\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id2522154\">Note that the <em>x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n\n<div id=\"fs-id1913210\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1832358\"><strong>Given an equation, graph by plotting points.<\/strong><\/p>\n\n<ol id=\"fs-id2302092\" type=\"1\">\n \t<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n \t<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make the graphing simpler.<\/li>\n \t<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\n \t<li>Plot the ordered pairs.<\/li>\n \t<li>Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_01_02\" class=\"textbox examples\">\n<div id=\"fs-id2931480\">\n<div id=\"fs-id2931196\">\n<h3>Graphing an Equation in Two Variables by Plotting Points<\/h3>\n<p id=\"fs-id1699173\">Graph the equation[latex]\\,y=-x+2\\,[\/latex]by plotting points.<\/p>\n\n<\/div>\n<div id=\"fs-id805727\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id805727\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id805727\"]\n<p id=\"fs-id1517317\">First, we construct a table similar to <a class=\"autogenerated-content\" href=\"#Table_02_01_02\">(Figure)<\/a>. Choose <em>x<\/em> values and calculate <em>y.<\/em><\/p>\n\n<table id=\"Table_02_01_02\" summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=-x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id2330968\">Now, plot the points. Connect them if they form a line. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_007\">(Figure)<\/a><\/p>\n\n<div id=\"Figure_02_01_007\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132929\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\"> <strong>Figure 6.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id833198\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_01\">\n<div id=\"fs-id2995656\">\n<p id=\"fs-id1752876\">Construct a table and graph the equation by plotting points:[latex]\\,y=\\frac{1}{2}x+2.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"3155135\"]Show Solution[\/reveal-answer][hidden-answer a=\"3155135\"]\n<table id=\"fs-id859388\" summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span id=\"fs-id2440549\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132932\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1901322\" class=\"bc-section section\">\n<h3>Graphing Equations with a Graphing Utility<\/h3>\n<p id=\"fs-id1722871\">Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style[latex]\\,y\\,[\/latex]=_____. The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.<\/p>\n<p id=\"fs-id1539399\">For example, the equation[latex]\\,y=2x-20\\,[\/latex]has been entered in the TI-84 Plus shown in <a class=\"autogenerated-content\" href=\"#Figure_02_01_09\">(Figure)<\/a><strong>a. <\/strong>In <a class=\"autogenerated-content\" href=\"#Figure_02_01_09\">(Figure)<\/a><strong>b, <\/strong>the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows[latex]\\,-10\\le x\\le 10,[\/latex]and[latex]\\,-10\\le y\\le 10.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_02_01_09\">(Figure)<\/a><strong>c<\/strong>.<\/p>\n\n<div id=\"Figure_02_01_09\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132937\/CNX_CAT_Figure_02_01_009abcN.jpg\" alt=\"This is an image of three side-by-side calculator screen captures. The first screen is the plot screen with the function y sub 1 equals two times x minus twenty. The second screen shows the plotted line on the coordinate plane. The third screen shows the window edit screen with the following settings: Xmin = -10; Xmax = 10; Xscl = 1; Ymin = -10; Ymax = 10; Yscl = 1; Xres = 1.\" width=\"731\" height=\"215\"> <strong> Figure 7. a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1416938\">By changing the window to show more of the positive <em>x-<\/em>axis and more of the negative <em>y-<\/em>axis, we have a much better view of the graph and the <em>x-<\/em> and <em>y-<\/em>intercepts. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_10\">(Figure)<\/a><strong>a<\/strong> and <a class=\"autogenerated-content\" href=\"#Figure_02_01_10\">(Figure)<\/a><strong>b.<\/strong><\/p>\n\n<div id=\"Figure_02_01_10\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132939\/CNX_CAT_Figure_02_01_010ab.jpg\" alt=\"This is an image of two side-by-side calculator screen captures. The first screen is the window edit screen with the following settings: Xmin = negative 5; Xmax = 15; Xscl = 1; Ymin = -30; Ymax = 10; Yscl = 1; Xres =1. The second screen shows the plot of the previous graph, but is more centered on the line.\" width=\"487\" height=\"213\"> <strong> Figure 8.<\/strong> a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.[\/caption]\n\n<div id=\"Example_02_01_03\" class=\"textbox examples\">\n<div id=\"fs-id2389098\">\n<div id=\"fs-id1336447\">\n<h3>Using a Graphing Utility to Graph an Equation<\/h3>\n<p id=\"fs-id1787357\">Use a graphing utility to graph the equation:[latex]\\,y=-\\frac{2}{3}x-\\frac{4}{3}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1513712\"]Show Solution[\/reveal-answer][hidden-answer a=\"1513712\"]\n<p id=\"fs-id781084\">Enter the equation in the <em>y=<\/em> function of the calculator. Set the window settings so that both the <em>x-<\/em> and <em>y-<\/em> intercepts are showing in the window. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_011\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_02_01_011\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132943\/CNX_CAT_Figure_02_01_011.jpg\" alt=\"This image is of a line graph on an x, y coordinate plane. The x-axis has numbers that range from negative 3 to 4. The y-axis has numbers that range from negative 3 to 3. The function y = -2x\/3 + 4\/3 is plotted.\" width=\"487\" height=\"343\"> <strong>Figure 9.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1340475\" class=\"bc-section section\">\n<h3>Finding <em>x-<\/em>intercepts and <em>y-<\/em>intercepts<\/h3>\n<p id=\"fs-id2503271\">The intercepts of a graph are points at which the graph crosses the axes. The <em>x-<\/em>intercept is the point at which the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <em>y-<\/em>intercept is the point at which the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p id=\"fs-id1448434\">To determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation[latex]\\,y=3x-1.[\/latex]<\/p>\n<p id=\"fs-id1493312\">To find the <em>x-<\/em>intercept, set[latex]\\,y=0.[\/latex]<\/p>\n\n<div id=\"fs-id3064821\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,y=3x-1\\hfill &amp; \\hfill \\\\ \\,0=3x-1\\hfill &amp; \\hfill \\\\ \\,1=3x\\hfill &amp; \\hfill \\\\ \\frac{1}{3}=x\\hfill &amp; \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill &amp; x\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2905820\">To find the <em>y-<\/em>intercept, set[latex]\\,x=0.[\/latex]<\/p>\n\n<div id=\"fs-id1798574\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=3x-1\\hfill \\\\ y=3\\left(0\\right)-1\\hfill \\\\ y=-1\\hfill \\\\ \\left(0,-1\\right)\\phantom{\\rule{3em}{0ex}}y\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1730534\">We can confirm that our results make sense by observing a graph of the equation as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_012\">(Figure)<\/a>. Notice that the graph crosses the axes where we predicted it would.<\/p>\n\n<div id=\"Figure_02_01_012\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132956\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\"> <strong>Figure 10.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1780189\" class=\"textbox key-takeaways\">\n<h3>Given an equation, find the intercepts.<\/h3>\n<ol id=\"fs-id2294904\" type=\"1\">\n \t<li>Find the <em>x<\/em>-intercept by setting[latex]\\,y=0\\,[\/latex]and solving for[latex]\\,x.[\/latex]<\/li>\n \t<li>Find the <em>y-<\/em>intercept by setting[latex]\\,x=0\\,[\/latex]and solving for[latex]\\,y.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1689084\">\n<div id=\"fs-id1419163\">\n<h3>Finding the Intercepts of the Given Equation<\/h3>\nFind the intercepts of the equation[latex]\\,y=-3x-4.\\,[\/latex]Then sketch the graph using only the intercepts.\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1514793\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1514793\"]\n<p id=\"fs-id1514793\">Set[latex]\\,y=0\\,[\/latex]to find the <em>x-<\/em>intercept.<\/p>\n\n<div id=\"fs-id2293732\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\phantom{\\rule{1em}{0ex}}y=-3x-4\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}0=-3x-4\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)\\phantom{\\rule{3em}{0ex}}x\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1560533\">Set[latex]\\,x=0\\,[\/latex]to find the <em>y-<\/em>intercept.<\/p>\n\n<div id=\"fs-id2803056\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=-3x-4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)\\phantom{\\rule{3.5em}{0ex}}y\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\nPlot both points, and draw a line passing through them as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_013\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133002\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\"> <strong>Figure 11.<\/strong>[\/caption]\n<p id=\"fs-id2504052\"><span id=\"fs-id2433829\">[\/hidden-answer]<\/span><span id=\"fs-id2433829\"><\/span><span id=\"fs-id2433829\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1518804\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_02\">\n<div id=\"fs-id2437906\">\n<p id=\"fs-id1972959\">Find the intercepts of the equation and sketch the graph:[latex]\\,y=-\\frac{3}{4}x+3.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1447496\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1447496\"]\n<p id=\"fs-id1447496\"><em>x<\/em>-intercept is[latex]\\,\\left(4,0\\right);[\/latex]<em>y-<\/em>intercept is[latex]\\,\\left(0,3\\right).[\/latex]<\/p>\n<span id=\"fs-id1297236\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133035\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1280821\" class=\"bc-section section\">\n<h3>Using the Distance Formula<\/h3>\n<p id=\"fs-id1277804\">Derived from the <span class=\"no-emphasis\">Pythagorean Theorem<\/span>, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,[latex]\\,{a}^{2}+{b}^{2}={c}^{2},[\/latex]is based on a right triangle where <em>a <\/em>and <em>b<\/em> are the lengths of the legs adjacent to the right angle, and <em>c<\/em> is the length of the hypotenuse. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_015\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_02_01_015\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133041\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1151919\">The relationship of sides[latex]\\,|{x}_{2}-{x}_{1}|\\,[\/latex]and[latex]\\,|{y}_{2}-{y}_{1}|\\,[\/latex]to side <em>d<\/em> is the same as that of sides <em>a <\/em>and <em>b <\/em>to side <em>c.<\/em> We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example,[latex]\\,|-3|=3.\\,[\/latex]) The symbols[latex]\\,|{x}_{2}-{x}_{1}|\\,[\/latex]and[latex]\\,|{y}_{2}-{y}_{1}|\\,[\/latex]indicate that the lengths of the sides of the triangle are positive. To find the length <em>c<\/em>, take the square root of both sides of the Pythagorean Theorem.<\/p>\n\n<div id=\"fs-id2730208\" class=\"unnumbered aligncenter\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\to c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/div>\n<p id=\"fs-id2666328\">It follows that the distance formula is given as<\/p>\n\n<div id=\"fs-id1188836\" class=\"unnumbered aligncenter\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<p id=\"fs-id1832560\">We do not have to use the absolute value symbols in this definition because any number squared is positive.<\/p>\n\n<div id=\"fs-id1521890\" class=\"textbox key-takeaways\">\n<h3>The Distance Formula<\/h3>\n<p id=\"eip-id2982180\">Given endpoints[latex]\\,\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and[latex]\\,\\left({x}_{2},{y}_{2}\\right),[\/latex]the distance between two points is given by<\/p>\n\n<div id=\"fs-id2933949\" class=\"unnumbered aligncenter\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_02_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1203380\">\n<div id=\"fs-id1204935\">\n<h3>Finding the Distance between Two Points<\/h3>\n<p id=\"fs-id1418038\">Find the distance between the points[latex]\\,\\left(-3,-1\\right)\\,[\/latex]and[latex]\\,\\left(2,3\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3223093\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3223093\"]\n<p id=\"fs-id3223093\">Let us first look at the graph of the two points. Connect the points to form a right triangle as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_016\">(Figure)<\/a><strong>.<\/strong><\/p>\n\n<div id=\"Figure_02_01_016\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133043\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1211736\">Then, calculate the length of <em>d <\/em>using the distance formula.<\/p>\n\n<div id=\"fs-id1839331\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{25+16}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{41}\\hfill \\end{array}\\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1574404\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_03\">\n<div id=\"fs-id1938614\">\n<p id=\"fs-id1938615\">Find the distance between two points:[latex]\\,\\left(1,4\\right)\\,[\/latex]and[latex]\\,\\left(11,9\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2667607\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2667607\"]\n<p id=\"fs-id2667607\">[latex]\\sqrt{125}=5\\sqrt{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_01_06\" class=\"textbox examples\">\n<div id=\"fs-id2503622\">\n<div id=\"fs-id1338610\">\n<h3>Finding the Distance between Two Locations<\/h3>\n<p id=\"fs-id2522450\">Let\u2019s return to the situation introduced at the beginning of this section.<\/p>\n<p id=\"fs-id2640218\">Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in <a class=\"autogenerated-content\" href=\"#Figure_02_01_001\">(Figure)<\/a>. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2522863\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2522863\"]\n<p id=\"fs-id2522863\">The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at[latex]\\,\\left(1,1\\right).\\,[\/latex]The next stop is 5 blocks to the east, so it is at[latex]\\,\\left(5,1\\right).\\,[\/latex]After that, she traveled 3 blocks east and 2 blocks north to[latex]\\,\\left(8,3\\right).\\,[\/latex]Lastly, she traveled 4 blocks north to[latex]\\,\\left(8,7\\right).\\,[\/latex]We can label these points on the grid as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_017\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_02_01_017\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133101\/CNX_CAT_Figure_02_01_017.jpg\" alt=\"This is an image of a road map of a city. The point (1, 1) is on North Avenue and Bertau Avenue. The point (5, 1) is on North Avenue and Wolf Road. The point (8, 3) is on Mannheim Road and McLean Street. The point (8, 7) is on Mannheim Road and Schiller Avenue.\" width=\"731\" height=\"480\"> <strong>Figure 14.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id2777115\">Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.<\/p>\n\n<ul id=\"fs-id1688170\">\n \t<li>From her starting location to her first stop at[latex]\\,\\left(1,1\\right),[\/latex]Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.<\/li>\n \t<li>Her second stop is at[latex]\\,\\left(5,1\\right).\\,[\/latex]So from[latex]\\,\\left(1,1\\right)\\,[\/latex]to[latex]\\,\\left(5,1\\right),[\/latex]Tracie drove east 4,000 feet.<\/li>\n \t<li>Her third stop is at[latex]\\,\\left(8,3\\right).\\,[\/latex]There are a number of routes from[latex]\\,\\left(5,1\\right)\\,[\/latex]to[latex]\\,\\left(8,3\\right).\\,[\/latex]Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let\u2019s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\n \t<li>Tracie\u2019s final stop is at[latex]\\,\\left(8,7\\right).\\,[\/latex]This is a straight drive north from[latex]\\,\\left(8,3\\right)\\,[\/latex]for a total of 4,000 feet.<\/li>\n<\/ul>\n<p id=\"fs-id1688205\">Next, we will add the distances listed in <a class=\"autogenerated-content\" href=\"#Table_02_01_03\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_02_01_03\" summary=\"A table with 6 rows and 2 columns. The entries in the first row are: From\/To and Number of Feet Driven. The entries in the second row are: (0, 0) to (1, 1) and 2,000. The entries in the third row are: (1, 1) to (5, 1) and 4,000. The entries in the fourth row are: (5, 1) to (8, 3) and 5,000. The entries in the fourth row are: (8, 3) to (8, 7) and 4,000. The entries in the sixth row are: Total and 15,000.\">\n<thead>\n<tr>\n<th>From\/To<\/th>\n<th>Number of Feet Driven<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(0,0\\right)\\,[\/latex]to[latex]\\,\\left(1,1\\right)[\/latex]<\/td>\n<td>2,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(1,1\\right)\\,[\/latex]to[latex]\\left(5,1\\right)\\,[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,1\\right)\\,[\/latex]to[latex]\\,\\left(8,3\\right)[\/latex]<\/td>\n<td>5,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(8,3\\right)\\,[\/latex]to[latex]\\,\\left(8,7\\right)[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>15,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id2629595\">The total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points[latex]\\,\\left(0,0\\right)\\,[\/latex]and[latex]\\,\\left(8,7\\right).[\/latex]<\/p>\n\n<div id=\"fs-id2959388\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}d=\\sqrt{{\\left(8-0\\right)}^{2}+{\\left(7-0\\right)}^{2}}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{64+49}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{113}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=10.63\\text{ units}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1475568\">At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point[latex]\\,\\left(8,7\\right).\\,[\/latex]Perhaps you have heard the saying \u201cas the crow flies,\u201d which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2507035\" class=\"bc-section section\">\n<h3>Using the Midpoint Formula<\/h3>\n<p id=\"fs-id1151538\">When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment,[latex]\\,\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and[latex]\\,\\left({x}_{2},{y}_{2}\\right),[\/latex]the midpoint formula states how to find the coordinates of the midpoint[latex]\\,M.[\/latex]<\/p>\n\n<div id=\"fs-id1520430\" class=\"unnumbered aligncenter\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\n<p id=\"fs-id2837053\">A graphical view of a midpoint is shown in <a class=\"autogenerated-content\" href=\"#Figure_02_01_018\">(Figure)<\/a>. Notice that the line segments on either side of the midpoint are congruent.<\/p>\n\n<div id=\"Figure_02_01_018\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133109\/CNX_CAT_Figure_02_01_018.jpg\" alt=\"This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.\" width=\"487\" height=\"290\"> <strong>Figure 15.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"Example_02_01_07\" class=\"textbox examples\">\n<div id=\"fs-id3008576\">\n<div id=\"fs-id3008579\">\n<h3>Finding the Midpoint of the Line Segment<\/h3>\n<p id=\"fs-id1336692\">Find the midpoint of the line segment with the endpoints[latex]\\,\\left(7,-2\\right)\\,[\/latex]and[latex]\\,\\left(9,5\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1213113\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1213113\"]\n<p id=\"fs-id1213113\">Use the formula to find the midpoint of the line segment.<\/p>\n\n<div id=\"fs-id1926574\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\phantom{\\rule{6.5em}{0ex}}=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1537533\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_04\">\n<div id=\"fs-id2434160\">\n<p id=\"fs-id1824194\">Find the midpoint of the line segment with endpoints[latex]\\,\\left(-2,-1\\right)\\,[\/latex]and[latex]\\,\\left(-8,6\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3008559\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3008559\"]\n<p id=\"fs-id3008559\">[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_01_08\" class=\"textbox examples\">\n<div id=\"fs-id1977814\">\n<div id=\"fs-id1977817\">\n<h3>Finding the Center of a Circle<\/h3>\n<p id=\"fs-id1977163\">The diameter of a circle has endpoints[latex]\\,\\left(-1,-4\\right)\\,[\/latex]and[latex]\\,\\left(5,-4\\right).\\,[\/latex]Find the center of the circle.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1523294\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1523294\"]\n<p id=\"fs-id1523294\">The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.<\/p>\n\n<div id=\"fs-id2858239\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4-4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1539521\" class=\"precalculus media\">\n<p id=\"fs-id2435248\">Access these online resources for additional instruction and practice with the Cartesian coordinate system.<\/p>\n\n<ul id=\"fs-id2766408\">\n \t<li><a href=\"http:\/\/Openstaxcollege.org\/l\/coordplotpnts\">Plotting points on the coordinate plane<\/a><\/li>\n \t<li><a href=\"http:\/\/Openstaxcollege.org\/l\/xyintsgraph\">Find x and y intercepts based on the graph of a line<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id2721419\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1449332\">\n \t<li>We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the <em>x-<\/em>axis and displacement from the <em>y-<\/em>axis. See <a class=\"autogenerated-content\" href=\"#Example_02_01_01\">(Figure)<\/a>.<\/li>\n \t<li>An equation can be graphed in the plane by creating a table of values and plotting points. See <a class=\"autogenerated-content\" href=\"#Example_02_01_02\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form <em>y=<\/em>_____. See <a class=\"autogenerated-content\" href=\"#Example_02_01_03\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>Finding the <em>x- <\/em>and <em>y-<\/em>intercepts can define the graph of a line. These are the points where the graph crosses the axes. See <a class=\"autogenerated-content\" href=\"#Example_02_01_04\">(Figure)<\/a>.<\/li>\n \t<li>The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See <a class=\"autogenerated-content\" href=\"#Example_02_01_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_01_06\">(Figure)<\/a>.<\/li>\n \t<li>The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the <em>x<\/em>-coordinates and the sum of the <em>y<\/em>-coordinates of the endpoints by 2. See <a class=\"autogenerated-content\" href=\"#Example_02_01_07\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_01_08\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1553580\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id2496131\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1355431\">\n<div id=\"fs-id1355432\">\n<p id=\"fs-id1355433\">Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1543668\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1543668\"]\n<p id=\"fs-id1543668\">Answers may vary. Yes. It is possible for a point to be on the <em>x<\/em>-axis or on the <em>y<\/em>-axis and therefore is considered to NOT be in one of the quadrants.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2435397\">\n<div>\n<p id=\"fs-id1929270\">Describe the process for finding the <em>x-<\/em>intercept and the <em>y<\/em>-intercept of a graph algebraically.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1351774\">\n<div id=\"fs-id2682309\">\n<p id=\"fs-id2682310\">Describe in your own words what the <em>y<\/em>-intercept of a graph is.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1322043\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1322043\"]\n<p id=\"fs-id1322043\">The <em>y<\/em>-intercept is the point where the graph crosses the <em>y<\/em>-axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1823207\">\n<div id=\"fs-id1823208\">\n<p id=\"fs-id1823209\">When using the distance formula[latex]\\,d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}},[\/latex]explain the correct order of operations that are to be performed to obtain the correct answer.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3263952\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1789811\">For each of the following exercises, find the <em>x<\/em>-intercept and the <em>y<\/em>-intercept without graphing. Write the coordinates of each intercept.<\/p>\n\n<div id=\"fs-id3039754\">\n<div id=\"fs-id3039755\">\n<p id=\"fs-id2512845\">[latex]y=-3x+6[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1441309\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1441309\"]\n<p id=\"fs-id1441309\">The <em>x-<\/em>intercept is[latex]\\,\\left(2,0\\right)\\,[\/latex]and the <em>y<\/em>-intercept is[latex]\\,\\left(0,6\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1499169\">\n<div id=\"fs-id1499170\">\n<p id=\"fs-id1499171\">[latex]4y=2x-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1007678\">\n<div id=\"fs-id1591152\">\n<p id=\"fs-id1591153\">[latex]3x-2y=6[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1769437\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1769437\"]\n<p id=\"fs-id1769437\">The <em>x-<\/em>intercept is[latex]\\,\\left(2,0\\right)\\,[\/latex]and the <em>y<\/em>-intercept is[latex]\\,\\left(0,-3\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2948006\">\n<div id=\"fs-id2948007\">\n<p id=\"fs-id2948008\">[latex]4x-3=2y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1723142\">\n<div id=\"fs-id1723143\">\n<p id=\"fs-id1937401\">[latex]3x+8y=9[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2737079\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2737079\"]\n<p id=\"fs-id2737079\">The <em>x-<\/em>intercept is[latex]\\,\\left(3,0\\right)\\,[\/latex]and the <em>y<\/em>-intercept is[latex]\\,\\left(0,\\frac{9}{8}\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1197890\">\n<div id=\"fs-id1197891\">\n<p id=\"fs-id1197892\">[latex]2x-\\frac{2}{3}=\\frac{3}{4}y+3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id3207567\">For each of the following exercises, solve the equation for <em>y<\/em> in terms of <em>x<\/em>.<\/p>\n\n<div id=\"fs-id3042175\">\n<div id=\"fs-id3042176\">\n<p id=\"fs-id2486646\">[latex]4x+2y=8[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1830748\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1830748\"]\n<p id=\"fs-id1830748\">[latex]y=4-2x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1238095\">\n<div id=\"fs-id1238096\">\n<p id=\"fs-id1238097\">[latex]3x-2y=6[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2512516\">\n<div id=\"fs-id1441357\">\n<p id=\"fs-id1441358\">[latex]2x=5-3y[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1939378\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1939378\"]\n<p id=\"fs-id1939378\">[latex]y=\\frac{5-2x}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1467950\">\n<div id=\"fs-id2521131\">\n<p id=\"fs-id2521132\">[latex]x-2y=7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1278706\">\n<div id=\"fs-id1402803\">\n<p id=\"fs-id1402804\">[latex]5y+4=10x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1688278\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1688278\"]\n<p id=\"fs-id1688278\">[latex]y=2x-\\frac{4}{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2388766\">\n<div id=\"fs-id2388767\">\n<p id=\"fs-id2952969\">[latex]5x+2y=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1702384\">For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.<\/p>\n\n<div id=\"fs-id1798648\">\n<div id=\"fs-id1798650\">\n<p id=\"fs-id1402574\">[latex]\\left(-4,1\\right)\\,[\/latex]and[latex]\\,\\left(3,-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1520540\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1520540\"]\n<p id=\"fs-id1520540\">[latex]d=\\sqrt{74}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1762329\">\n<div id=\"fs-id2947043\">\n<p id=\"fs-id2947044\">[latex]\\left(2,-5\\right)\\,[\/latex]and[latex]\\,\\left(7,4\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1932412\">\n<div id=\"fs-id2266178\">\n<p id=\"fs-id2266179\">[latex]\\left(5,0\\right)\\,[\/latex]and[latex]\\,\\left(5,6\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2729922\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2729922\"]\n<p id=\"fs-id2729922\">[latex]d=\\sqrt{36}=6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1538085\">\n<div id=\"fs-id1538086\">\n<p id=\"fs-id1538087\">[latex]\\left(-4,3\\right)\\,[\/latex]and[latex]\\,\\left(10,3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2628482\">\n<div id=\"fs-id2628483\">\n<p id=\"fs-id2628484\">Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.<\/p>\n<p id=\"fs-id2644287\">[latex]\\left(19,12\\right)\\,[\/latex]and[latex]\\,\\left(41,71\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2411257\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2411257\"]\n<p id=\"fs-id2411257\">[latex]d\\approx 62.97[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2753833\">For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.<\/p>\n\n<div id=\"fs-id2389564\">\n<div id=\"fs-id2389565\">\n<p id=\"fs-id2389566\">[latex]\\left(-5,-6\\right)\\,[\/latex]and[latex]\\,\\left(4,2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1386855\">\n<div id=\"fs-id1386856\">\n<p id=\"fs-id1386857\">[latex]\\left(-1,1\\right)\\,[\/latex]and[latex]\\,\\left(7,-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1225500\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1225500\"]\n<p id=\"fs-id1225500\">[latex]\\left(3,\\frac{-3}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2425333\">\n<div id=\"fs-id2425334\">\n<p id=\"fs-id2425335\">[latex]\\left(-5,-3\\right)\\,[\/latex]and[latex]\\,\\left(-2,-8\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2736528\">\n<div id=\"fs-id2736529\">\n<p id=\"fs-id2736530\">[latex]\\left(0,7\\right)\\,[\/latex]and[latex]\\,\\left(4,-9\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2694213\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2694213\"]\n<p id=\"fs-id2694213\">[latex]\\left(2,-1\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1960033\">\n<div id=\"fs-id1960034\">\n<p id=\"fs-id2431256\">[latex]\\left(-43,17\\right)\\,[\/latex]and[latex]\\,\\left(23,-34\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2434980\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1919734\">For each of the following exercises, identify the information requested.<\/p>\n\n<div id=\"fs-id1940524\">\n<div id=\"fs-id1940525\">\n<p id=\"fs-id1422420\">What are the coordinates of the origin?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3162327\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3162327\"]\n<p id=\"fs-id3162327\">[latex]\\left(0,0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1965313\">\n<div id=\"fs-id1965314\">\n<p id=\"fs-id1965315\">If a point is located on the <em>y<\/em>-axis, what is the <em>x<\/em>-coordinate?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1575040\">\n<div id=\"fs-id1575041\">\n<p id=\"fs-id1575042\">If a point is located on the <em>x<\/em>-axis, what is the <em>y<\/em>-coordinate?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1583783\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1583783\"]\n<p id=\"fs-id1583783\">[latex]y=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1475798\">For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).<\/p>\n\n<div id=\"fs-id1182006\">\n<div id=\"fs-id1182007\">\n<p id=\"fs-id1182008\">[latex]\\left(4,1\\right)\\left(-2,-3\\right)\\left(5,0\\right)[\/latex]<\/p>\n<span id=\"fs-id2387110\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133120\/CNX_CAT_Figure_02_01_201.jpg\" alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1341660\">\n<div id=\"fs-id1341661\">\n<p id=\"fs-id1341662\">[latex]\\left(-1,2\\right)\\left(0,4\\right)\\left(2,1\\right)[\/latex]<\/p>\n<span id=\"fs-id3165041\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133120\/CNX_CAT_Figure_02_01_201.jpg\" alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\"><\/span>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1319659\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1319659\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133124\/CNX_CAT_Figure_02_01_203.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (0,4); (-1,2) and (2,1) are plotted and labeled.\">not collinear[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2440944\">\n<div id=\"fs-id2440946\">\n<p id=\"fs-id2440947\">[latex]\\left(-3,0\\right)\\left(-3,4\\right)\\left(-3,-3\\right)[\/latex]<\/p>\n<span id=\"fs-id2918881\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133120\/CNX_CAT_Figure_02_01_201.jpg\" alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2453414\">\n<div id=\"fs-id2453415\">\n<p id=\"fs-id2453416\">Name the coordinates of the points graphed.<\/p>\n<span id=\"fs-id1333042\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133131\/CNX_CAT_Figure_02_01_205.jpg\" alt=\"This is an image of an x, y coordinate plane where the x and y-axis range from negative 5 to 5. Three points are plotted: A, B, and C.\"><\/span>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3158703\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3158703\"]\n<p id=\"fs-id3158703\">[latex]\\left(-3,2\\right),\\left(1,3\\right),\\left(4,0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3155305\">\n<div id=\"fs-id3155306\">\n<p id=\"fs-id3155307\">Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.<\/p>\n<p id=\"fs-id1844064\">[latex]\\begin{array}{l}a.\\left(-3,-4\\right)\\\\ b.\\left(-5,0\\right)\\\\ c.\\left(1,-4\\right)\\\\ d.\\left(-2,7\\right)\\\\ e.\\left(0,-3\\right)\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1569563\">For each of the following exercises, construct a table and graph the equation by plotting at least three points.<\/p>\n\n<div id=\"fs-id1569567\">\n<div id=\"fs-id3182657\">\n<p id=\"fs-id3182658\">[latex]y=\\frac{1}{3}x+2[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1538490\"]Show Solution[\/reveal-answer][hidden-answer a=\"1538490\"]\n<table id=\"fs-id2801280\" summary=\"A table with 5 rows and 2 columns. The entries in the first row are: x and y. The entries in the second row are: negative 3 and 1. The entries in the third row are: 0 and 2. The entries in the fourth row are: 3 and 3. The entries in the fifth row are: 6 and 4.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span id=\"fs-id1920364\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133143\/CNX_CAT_Figure_02_01_206.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (-3, 1); (0, 2); (3, 3) and (6, 4) are plotted and labeled. A line runs through all these points.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3070032\">\n<div id=\"fs-id3070034\">\n<p id=\"fs-id3070035\">[latex]y=-3x+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1445099\">\n<div id=\"fs-id1477477\">\n<p id=\"fs-id1477478\">[latex]2y=x+3[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1764492\"]Show Solution[\/reveal-answer][hidden-answer a=\"1764492\"]\n<table id=\"fs-id1764494\" summary=\"A table with 4 rows and 2 columns. The entries in the first row are: x and y. The entries in the second row are: negative 3 and 0. The entries in the third row are: 0 and 1.5. The entries in the fourth row are: 3 and 3.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><em>x<\/em><\/td>\n<td><em>y<\/em><\/td>\n<\/tr>\n<tr>\n<td>\u20133<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>1.5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span id=\"fs-id1269392\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133151\/CNX_CAT_Figure_02_01_208.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (-3, 0); (0, 1.5) and (3, 3) are plotted and labeled. A line runs through all of these points.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1387762\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id2528941\">For each of the following exercises, find and plot the <em>x-<\/em> and <em>y<\/em>-intercepts, and graph the straight line based on those two points.<\/p>\n\n<div id=\"fs-id766182\">\n<div id=\"fs-id766183\">\n<p id=\"fs-id766184\">[latex]4x-3y=12[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3176745\">\n<div id=\"fs-id3176746\">\n<p id=\"fs-id3176748\">[latex]x-2y=8[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"3105867\"]Show Solution[\/reveal-answer][hidden-answer a=\"3105867\"]<span id=\"fs-id1333412\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133158\/CNX_CAT_Figure_02_01_210.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (8, 0) and (0, -4) are plotted and labeled. A line runs through both of these points.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1272882\">\n<div id=\"fs-id1272883\">\n<p id=\"fs-id1272884\">[latex]y-5=5x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1951937\">\n<div id=\"fs-id1951938\">\n<p id=\"fs-id1951940\">[latex]3y=-2x+6[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2385556\"]Show Solution[\/reveal-answer][hidden-answer a=\"2385556\"]<span id=\"fs-id2385560\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133207\/CNX_CAT_Figure_02_01_212.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (0, 2) and (3, 0) are plotted and labeled. A line runs through both of these points.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1517685\">\n<div id=\"fs-id1929175\">\n<p id=\"fs-id1929176\">[latex]y=\\frac{x-3}{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1919624\">For each of the following exercises, use the graph in the figure below.<\/p>\n<span id=\"fs-id1832449\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133214\/CNX_CAT_Figure_02_01_214.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (-3, 4) and (5, 2) are plotted. A line connects these two points.\"><\/span>\n<div id=\"fs-id1267899\">\n<div id=\"fs-id2432278\">\n<p id=\"fs-id2432279\">Find the distance between the two endpoints using the distance formula. Round to three decimal places.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2905920\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2905920\"]\n<p id=\"fs-id2905920\">[latex]d=8.246[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1895437\">\n<div id=\"fs-id1815372\">\n<p id=\"fs-id1815373\">Find the coordinates of the midpoint of the line segment connecting the two points.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1815376\">\n<div id=\"fs-id1418769\">\n<p id=\"fs-id1418770\">Find the distance that[latex]\\,\\left(-3,4\\right)\\,[\/latex]is from the origin.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1258397\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1258397\"]\n<p id=\"fs-id1258397\">[latex]d=5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1892569\">\n<div id=\"fs-id1892570\">\n<p id=\"fs-id1892571\">Find the distance that[latex]\\,\\left(5,2\\right)\\,[\/latex]is from the origin. Round to three decimal places.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2437516\">\n<div id=\"fs-id2437517\">\n<p id=\"fs-id1691144\">Which point is closer to the origin?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1691149\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1691149\"]\n<p id=\"fs-id1691149\">[latex]\\left(-3,4\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2437604\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1333818\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\n<p id=\"fs-id1333822\">After graphing it, use the 2<sup>nd<\/sup> CALC button and 1:value button, hit enter. At the lower part of the screen you will see \u201cx=\u201d and a blinking cursor. You may enter any number for <em>x<\/em> and it will display the <em>y<\/em> value for any <em>x<\/em> value you input. Use this and plug in <em>x<\/em> = 0, thus finding the <em>y<\/em>-intercept, for each of the following graphs.<\/p>\n\n<div id=\"fs-id1686604\">\n<div id=\"fs-id1686605\">\n<p id=\"fs-id1686606\">[latex]{\\text{Y}}_{1}=-2x+5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1181943\">\n<div id=\"fs-id1181944\">\n<p id=\"fs-id1181945\">[latex]{\\text{Y}}_{1}=\\frac{3x-8}{4}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2639345\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2639345\"]\n<p id=\"fs-id2639345\">[latex]x=0\\text{ }y=-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2800083\">\n<div id=\"fs-id1421792\">\n<p id=\"fs-id1421793\">[latex]{\\text{Y}}_{1}=\\frac{x+5}{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2785067\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\n<p id=\"fs-id2384791\">After graphing it, use the 2<sup>nd<\/sup> CALC button and 2:zero button, hit enter. At the lower part of the screen you will see \u201cleft bound?\u201d and a blinking cursor on the graph of the line. Move this cursor to the left of the <em>x<\/em>-intercept, hit ENTER. Now it says \u201cright bound?\u201d Move the cursor to the right of the <em>x<\/em>-intercept, hit enter. Now it says \u201cguess?\u201d Move your cursor to the left somewhere in between the left and right bound near the <em>x<\/em>-intercept. Hit enter. At the bottom of your screen it will display the coordinates of the <em>x-<\/em>intercept or the \u201czero\u201d to the <em>y<\/em>-value. Use this to find the <em>x<\/em>-intercept.<\/p>\n<p id=\"fs-id2016100\">Note: With linear\/straight line functions the zero is not really a \u201cguess,\u201d but it is necessary to enter a \u201cguess\u201d so it will search and find the exact <em>x<\/em>-intercept between your right and left boundaries. With other types of functions (more than one <em>x<\/em>-intercept), they may be irrational numbers so \u201cguess\u201d is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.<\/p>\n\n<div id=\"fs-id1425403\">\n<div id=\"fs-id1425404\">\n<p id=\"fs-id1425405\">[latex]{\\text{Y}}_{1}=-8x+6[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2016707\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2016707\"]\n<p id=\"fs-id2016707\">[latex]x=0.75\\text{ }y=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1845247\">\n<div id=\"fs-id1845248\">\n<p id=\"fs-id1845249\">[latex]{\\text{Y}}_{1}=4x-7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2387375\">\n<div id=\"fs-id2387376\">\n<p id=\"fs-id1673534\">[latex]{\\text{Y}}_{1}=\\frac{3x+5}{4}\\,[\/latex]Round your answer to the nearest thousandth.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2781155\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2781155\"]\n<p id=\"fs-id2781155\">[latex]x=-1.667\\text{ }y=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1932603\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1932608\">\n<div id=\"fs-id1333979\">\n<p id=\"fs-id1333980\">A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2513497\">\n<div id=\"fs-id2513498\">\n<p id=\"fs-id2513499\">If the road was made in the previous exercise, how much shorter would the man\u2019s one-way trip be every day?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1513803\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1513803\"]\n<p id=\"fs-id1513803\">[latex]\\text{15}\\text{\u221211}.\\text{2 }=\\text{ 3}.8\\,[\/latex]mi shorter<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2508338\">\n<div id=\"fs-id2508339\">\n<p id=\"fs-id1803338\">Given these four points:[latex]\\,A\\left(1,3\\right),\\text{}B\\left(-3,5\\right),\\text{}C\\left(4,7\\right),\\text{ and }D\\left(5,-4\\right),[\/latex]find the coordinates of the midpoint of line segments[latex]\\,\\overline{\\text{AB}}\\,[\/latex]and[latex]\\,\\overline{\\text{CD}}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1343510\">\n<div id=\"fs-id1343511\">\n<p id=\"fs-id1343512\">After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1960062\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1960062\"]\n<p id=\"fs-id1960062\">[latex]\\text{6}.0\\text{42}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1312332\">\n<div id=\"fs-id1312333\">\n<p id=\"fs-id3207842\">Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.<\/p>\n<span id=\"fs-id3207847\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133216\/CNX_CAT_Figure_02_01_215.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 12 to 12. The points (-6, 5); (10, 5); (-6, -1) and (10, -1) are plotted and labeled. These points are connected to form a rectangle. Dotted lines extend from each corner point to their opposite point.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1228174\">\n<div id=\"fs-id1228176\">\n<p id=\"fs-id1228177\">In the previous exercise, find the coordinates of the midpoint for each diagonal.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1542004\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1542004\"]\n<p id=\"fs-id1542004\">Midpoint of each diagonal is the same point[latex]\\,\\left(2,2\\right).\\,[\/latex]Note this is a characteristic of rectangles, but not other quadrilaterals.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2454474\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id2722618\">\n<div id=\"fs-id2722619\">\n<p id=\"fs-id2722620\">The coordinates on a map for San Francisco are[latex]\\,\\left(53,17\\right)\\,[\/latex]and those for Sacramento are[latex]\\,\\left(123,78\\right).\\,[\/latex]Note that coordinates represent miles. Find the distance between the cities to the nearest mile.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1511688\">\n<div id=\"fs-id1511690\">\n<p id=\"fs-id1511691\">If San Jose\u2019s coordinates are[latex]\\,\\left(76,-12\\right),[\/latex]where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2700244\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2700244\"]\n<p id=\"fs-id2700244\">[latex]\\text{37}\\,[\/latex]mi<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2925632\">\n<div id=\"fs-id2925633\">\n<p id=\"fs-id2925634\">A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were[latex]\\,\\left(49,64\\right).\\,[\/latex]One rescue boat is at the coordinates[latex]\\,\\left(60,82\\right)\\,[\/latex]and a second Coast Guard craft is at coordinates[latex]\\,\\left(58,47\\right).\\,[\/latex]Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1551836\">\n<div id=\"fs-id1551837\">\n<p id=\"fs-id1551838\">A man on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?<\/p>\n<span id=\"fs-id1477367\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133228\/CNX_CAT_Figure_02_01_217.jpg\" alt=\"A right triangle with its bottom left point sitting on the point (0,0). The upper right hand corner is labeled (20,50). The base has a length of 20 units and the triangle has a height of 50 units.\"><\/span>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1783293\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1783293\"]\n<p id=\"fs-id1783293\">54 ft<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1717974\">\n<div id=\"fs-id1717976\">\n<p id=\"fs-id1717977\">If we rent a truck and pay a $75\/day fee plus $.20 for every mile we travel, write a linear equation that would express the total cost[latex]\\,y,[\/latex]using[latex]\\,x\\,[\/latex]to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1890952\">\n \t<dt>Cartesian coordinate system<\/dt>\n \t<dd id=\"fs-id1402320\">a grid system designed with perpendicular axes invented by Ren\u00e9 Descartes<\/dd>\n<\/dl>\n<dl id=\"fs-id1402323\">\n \t<dt>distance formula<\/dt>\n \t<dd id=\"fs-id1574693\">a formula that can be used to find the length of a line segment if the endpoints are known<\/dd>\n<\/dl>\n<dl id=\"fs-id1574696\">\n \t<dt>equation in two variables<\/dt>\n \t<dd id=\"fs-id1277856\">a mathematical statement, typically written in <em>x <\/em>and <em>y<\/em>, in which two expressions are equal<\/dd>\n<\/dl>\n<dl id=\"fs-id2508809\">\n \t<dt>graph in two variables<\/dt>\n \t<dd id=\"fs-id2508812\">the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1277986\">\n \t<dt>intercepts<\/dt>\n \t<dd id=\"fs-id1277990\">the points at which the graph of an equation crosses the <em>x<\/em>-axis and the <em>y<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1337623\">\n \t<dt>midpoint formula<\/dt>\n \t<dd id=\"fs-id3082583\">a formula to find the point that divides a line segment into two parts of equal length<\/dd>\n<\/dl>\n<dl id=\"fs-id3082586\">\n \t<dt>ordered pair<\/dt>\n \t<dd id=\"fs-id1223328\">a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, [latex]\\,\\left(x,y\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id2502726\">\n \t<dt>origin<\/dt>\n \t<dd id=\"fs-id1534441\">the point where the two axes cross in the center of the plane, described by the ordered pair[latex]\\,\\left(0,0\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id3067555\">\n \t<dt>quadrant<\/dt>\n \t<dd id=\"fs-id2875924\">one quarter of the coordinate plane, created when the axes divide the plane into four sections<\/dd>\n<\/dl>\n<dl id=\"fs-id2875927\">\n \t<dt><em>x<\/em>-axis<\/dt>\n \t<dd id=\"fs-id2389105\">the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right<\/dd>\n<\/dl>\n<dl id=\"fs-id1504116\">\n \t<dt><em>x-<\/em>coordinate<\/dt>\n \t<dd id=\"fs-id2302378\">the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1227889\">\n \t<dt><em>x-<\/em>intercept<\/dt>\n \t<dd id=\"fs-id1278364\">the point where a graph intersects the <em>x-<\/em>axis; an ordered pair with a <em>y<\/em>-coordinate of zero<\/dd>\n<\/dl>\n<dl id=\"fs-id1257845\">\n \t<dt><em>y<\/em>-axis<\/dt>\n \t<dd id=\"fs-id815935\">the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top<\/dd>\n<\/dl>\n<dl id=\"fs-id2629843\">\n \t<dt><em>y-<\/em>coordinate<\/dt>\n \t<dd id=\"fs-id1521234\">the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id2364783\">\n \t<dt><em>y<\/em>-intercept<\/dt>\n \t<dd id=\"fs-id1575741\">a point where a graph intercepts the <em>y-<\/em>axis; an ordered pair with an <em>x<\/em>-coordinate of zero<\/dd>\n<\/dl>\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>Plot ordered pairs in a Cartesian coordinate system.<\/li>\n<li>Graph equations by plotting points.<\/li>\n<li>Graph equations with a graphing utility.<\/li>\n<li>Find [latex]x[\/latex]-intercepts and [latex]y[\/latex]-intercepts.<\/li>\n<li>Use the distance formula.<\/li>\n<li>Use the midpoint formula.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_02_01_001\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132911\/CNX_CAT_Figure_02_01_001.jpg\" alt=\"Road map of a city with street names on an x, y coordinate grid. Various points are marked in red on the grid lines indicating different locations on the map.\" width=\"731\" height=\"480\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id2906377\">Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in <a class=\"autogenerated-content\" href=\"#Figure_02_01_001\">(Figure)<\/a>. Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.<\/p>\n<div id=\"fs-id1392675\" class=\"bc-section section\">\n<h3>Plotting Ordered Pairs in the Cartesian Coordinate System<\/h3>\n<p id=\"fs-id2500615\">An old story describes how seventeenth-century philosopher\/mathematician Ren\u00e9 Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly\u2019s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers\u2014the displacement from the horizontal axis and the displacement from the vertical axis.<\/p>\n<p id=\"fs-id1960277\">While there is evidence that ideas similar to Descartes\u2019 grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the <em>x-<\/em>axis and the vertical axis the <em>y-<\/em>axis.<\/p>\n<p id=\"fs-id1167648\">The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the <em>x<\/em>-axis and the <em>y<\/em>-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in <a class=\"autogenerated-content\" href=\"#Figure_02_01_002\">(Figure)<\/a><\/p>\n<div id=\"Figure_02_01_002\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132913\/CNX_CAT_Figure_02_01_002.jpg\" alt=\"This is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1423341\">The center of the plane is the point at which the two axes cross. It is known as the origin, or point[latex]\\left(0,0\\right).[\/latex]From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the <em>x-<\/em>axis and up the <em>y-<\/em>axis; decreasing, negative numbers to the left on the <em>x-<\/em>axis and down the <em>y-<\/em>axis. The axes extend to positive and negative infinity as shown by the arrowheads in <a class=\"autogenerated-content\" href=\"#Figure_02_01_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_02_01_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132919\/CNX_CAT_Figure_02_01_003.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1787344\">Each point in the plane is identified by its <em>x-<\/em>coordinate, or horizontal displacement from the origin, and its <em>y-<\/em>coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form[latex]\\,\\left(x,y\\right).\\,[\/latex]An ordered pair is also known as a coordinate pair because it consists of <em>x-<\/em> and <em>y<\/em>-coordinates. For example, we can represent the point[latex]\\,\\left(3,-1\\right)\\,[\/latex]in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_004\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_02_01_004\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132921\/CNX_CAT_Figure_02_01_004.jpg\" alt=\"This is an image of an x, y coordinate plane. The x and y axis range from negative 5 to 5. The point (3, -1) is labeled. An arrow extends rightward from the origin 3 units and another arrow extends downward one unit from the end of that arrow to the point.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id3155322\">When dividing the axes into equally spaced increments, note that the <em>x-<\/em>axis may be considered separately from the <em>y-<\/em>axis. In other words, while the <em>x-<\/em>axis may be divided and labeled according to consecutive integers, the <em>y-<\/em>axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.<\/p>\n<div id=\"fs-id2782510\" class=\"textbox key-takeaways\">\n<h3>Cartesian Coordinate System<\/h3>\n<p id=\"fs-id1400039\">A two-dimensional plane where the<\/p>\n<ul id=\"fs-id573737\">\n<li><em>x<\/em>-axis is the horizontal axis<\/li>\n<li><em>y<\/em>-axis is the vertical axis<\/li>\n<\/ul>\n<p id=\"fs-id3085633\">A point in the plane is defined as an ordered pair,[latex]\\,\\left(x,y\\right),[\/latex]such that <em>x <\/em>is determined by its horizontal distance from the origin and <em>y <\/em>is determined by its vertical distance from the origin.<\/p>\n<\/div>\n<div id=\"Example_02_01_01\" class=\"textbox examples\">\n<div id=\"fs-id2270902\">\n<div id=\"fs-id1931516\">\n<h3>Plotting Points in a Rectangular Coordinate System<\/h3>\n<p id=\"fs-id1324875\">Plot the points[latex]\\,\\left(-2,4\\right),[\/latex][latex]\\left(3,3\\right),[\/latex]and[latex]\\,\\left(0,-3\\right)\\,[\/latex]in the plane.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1718016\">To plot the point[latex]\\,\\left(-2,4\\right),[\/latex]begin at the origin. The <em>x<\/em>-coordinate is \u20132, so move two units to the left. The <em>y<\/em>-coordinate is 4, so then move four units up in the positive <em>y <\/em>direction.<\/p>\n<p id=\"fs-id2958885\">To plot the point[latex]\\,\\left(3,3\\right),[\/latex]begin again at the origin. The <em>x<\/em>-coordinate is 3, so move three units to the right. The <em>y<\/em>-coordinate is also 3, so move three units up in the positive <em>y <\/em>direction.<\/p>\n<p id=\"fs-id2433014\">To plot the point[latex]\\,\\left(0,-3\\right),[\/latex]begin again at the origin. The <em>x<\/em>-coordinate is 0. This tells us not to move in either direction along the <em>x<\/em>-axis. The <em>y<\/em>-coordinate is \u20133, so move three units down in the negative <em>y<\/em> direction. See the graph in <a class=\"autogenerated-content\" href=\"#Figure_02_01_005\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_02_01_005\" class=\"small wp-caption aligncenter\"><span id=\"fs-id2187492\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132923\/CNX_CAT_Figure_02_01_005.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5. The points (-2, 4); (3, 3); and (0, -3) are labeled. Arrows extend from the origin to the points.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1391320\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1215275\">Note that when either coordinate is zero, the point must be on an axis. If the <em>x<\/em>-coordinate is zero, the point is on the <em>y<\/em>-axis. If the <em>y<\/em>-coordinate is zero, the point is on the <em>x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1940663\" class=\"bc-section section\">\n<h3>Graphing Equations by Plotting Points<\/h3>\n<p id=\"fs-id1951777\">We can plot a set of points to represent an equation. When such an equation contains both an <em>x <\/em>variable and a <em>y <\/em>variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.<\/p>\n<p id=\"fs-id1811199\">Suppose we want to graph the equation[latex]\\,y=2x-1.\\,[\/latex]We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of <em>y<\/em>. Each pair of <em>x<\/em>&#8211; and <em>y<\/em>-values is an ordered pair that can be plotted. <a class=\"autogenerated-content\" href=\"#Table_02_01_01\">(Figure)<\/a> lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.<\/p>\n<table id=\"Table_02_01_01\" summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=2x-1[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1386116\">We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_006\">(Figure)<\/a><strong>. <\/strong>This is not true for all equations.<\/p>\n<div id=\"Figure_02_01_006\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132927\/CNX_CAT_Figure_02_01_006.jpg\" alt=\"This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).\" width=\"731\" height=\"669\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id2522154\">Note that the <em>x-<\/em>values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of <em>x<\/em> to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.<\/p>\n<div id=\"fs-id1913210\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1832358\"><strong>Given an equation, graph by plotting points.<\/strong><\/p>\n<ol id=\"fs-id2302092\" type=\"1\">\n<li>Make a table with one column labeled <em>x<\/em>, a second column labeled with the equation, and a third column listing the resulting ordered pairs.<\/li>\n<li>Enter <em>x-<\/em>values down the first column using positive and negative values. Selecting the <em>x-<\/em>values in numerical order will make the graphing simpler.<\/li>\n<li>Select <em>x-<\/em>values that will yield <em>y-<\/em>values with little effort, preferably ones that can be calculated mentally.<\/li>\n<li>Plot the ordered pairs.<\/li>\n<li>Connect the points if they form a line.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_01_02\" class=\"textbox examples\">\n<div id=\"fs-id2931480\">\n<div id=\"fs-id2931196\">\n<h3>Graphing an Equation in Two Variables by Plotting Points<\/h3>\n<p id=\"fs-id1699173\">Graph the equation[latex]\\,y=-x+2\\,[\/latex]by plotting points.<\/p>\n<\/div>\n<div id=\"fs-id805727\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1517317\">First, we construct a table similar to <a class=\"autogenerated-content\" href=\"#Table_02_01_02\">(Figure)<\/a>. Choose <em>x<\/em> values and calculate <em>y.<\/em><\/p>\n<table id=\"Table_02_01_02\" summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=-x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id2330968\">Now, plot the points. Connect them if they form a line. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_007\">(Figure)<\/a><\/p>\n<div id=\"Figure_02_01_007\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132929\/CNX_CAT_Figure_02_01_007.jpg\" alt=\"This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).\" width=\"731\" height=\"556\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id833198\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_01\">\n<div id=\"fs-id2995656\">\n<p id=\"fs-id1752876\">Construct a table and graph the equation by plotting points:[latex]\\,y=\\frac{1}{2}x+2.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id859388\" summary=\"The table shows 6 rows and 3 columns. The entries in the first row are: x; y = x divided by 2 plus 2, (x,y). The entries in the second row are: negative 2; y = (negative 2) divided by 2 plus 2 = 1; (-2, 1). The entries in the third row are: negative 1; y = (negative 1) divided by 2 plus 2 = 3\/2; (-1,3\/2). The entries in the fourth row are: 0; y = (0)\/2 + 2 = 2; (0,2). The entries in the fifth row are: 1; y = (1)\/2 + 2 = 5\/2; (1,5\/2). The entries in the sixth row are: 2; y = (2)\/2 + 2 = 3; (2,3).\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span id=\"fs-id2440549\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132932\/CNX_CAT_Figure_02_01_008.jpg\" alt=\"This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3\/2); (0, 2); (1, 5\/2); and (2, 3).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1901322\" class=\"bc-section section\">\n<h3>Graphing Equations with a Graphing Utility<\/h3>\n<p id=\"fs-id1722871\">Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style[latex]\\,y\\,[\/latex]=_____. The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.<\/p>\n<p id=\"fs-id1539399\">For example, the equation[latex]\\,y=2x-20\\,[\/latex]has been entered in the TI-84 Plus shown in <a class=\"autogenerated-content\" href=\"#Figure_02_01_09\">(Figure)<\/a><strong>a. <\/strong>In <a class=\"autogenerated-content\" href=\"#Figure_02_01_09\">(Figure)<\/a><strong>b, <\/strong>the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows[latex]\\,-10\\le x\\le 10,[\/latex]and[latex]\\,-10\\le y\\le 10.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_02_01_09\">(Figure)<\/a><strong>c<\/strong>.<\/p>\n<div id=\"Figure_02_01_09\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132937\/CNX_CAT_Figure_02_01_009abcN.jpg\" alt=\"This is an image of three side-by-side calculator screen captures. The first screen is the plot screen with the function y sub 1 equals two times x minus twenty. The second screen shows the plotted line on the coordinate plane. The third screen shows the window edit screen with the following settings: Xmin = -10; Xmax = 10; Xscl = 1; Ymin = -10; Ymax = 10; Yscl = 1; Xres = 1.\" width=\"731\" height=\"215\" \/><figcaption class=\"wp-caption-text\"><strong> Figure 7. a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1416938\">By changing the window to show more of the positive <em>x-<\/em>axis and more of the negative <em>y-<\/em>axis, we have a much better view of the graph and the <em>x-<\/em> and <em>y-<\/em>intercepts. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_10\">(Figure)<\/a><strong>a<\/strong> and <a class=\"autogenerated-content\" href=\"#Figure_02_01_10\">(Figure)<\/a><strong>b.<\/strong><\/p>\n<div id=\"Figure_02_01_10\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132939\/CNX_CAT_Figure_02_01_010ab.jpg\" alt=\"This is an image of two side-by-side calculator screen captures. The first screen is the window edit screen with the following settings: Xmin = negative 5; Xmax = 15; Xscl = 1; Ymin = -30; Ymax = 10; Yscl = 1; Xres =1. The second screen shows the plot of the previous graph, but is more centered on the line.\" width=\"487\" height=\"213\" \/><figcaption class=\"wp-caption-text\"><strong> Figure 8.<\/strong> a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.<\/figcaption><\/figure>\n<div id=\"Example_02_01_03\" class=\"textbox examples\">\n<div id=\"fs-id2389098\">\n<div id=\"fs-id1336447\">\n<h3>Using a Graphing Utility to Graph an Equation<\/h3>\n<p id=\"fs-id1787357\">Use a graphing utility to graph the equation:[latex]\\,y=-\\frac{2}{3}x-\\frac{4}{3}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id781084\">Enter the equation in the <em>y=<\/em> function of the calculator. Set the window settings so that both the <em>x-<\/em> and <em>y-<\/em> intercepts are showing in the window. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_011\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_02_01_011\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132943\/CNX_CAT_Figure_02_01_011.jpg\" alt=\"This image is of a line graph on an x, y coordinate plane. The x-axis has numbers that range from negative 3 to 4. The y-axis has numbers that range from negative 3 to 3. The function y = -2x\/3 + 4\/3 is plotted.\" width=\"487\" height=\"343\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1340475\" class=\"bc-section section\">\n<h3>Finding <em>x-<\/em>intercepts and <em>y-<\/em>intercepts<\/h3>\n<p id=\"fs-id2503271\">The intercepts of a graph are points at which the graph crosses the axes. The <em>x-<\/em>intercept is the point at which the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <em>y-<\/em>intercept is the point at which the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p id=\"fs-id1448434\">To determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation[latex]\\,y=3x-1.[\/latex]<\/p>\n<p id=\"fs-id1493312\">To find the <em>x-<\/em>intercept, set[latex]\\,y=0.[\/latex]<\/p>\n<div id=\"fs-id3064821\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,y=3x-1\\hfill & \\hfill \\\\ \\,0=3x-1\\hfill & \\hfill \\\\ \\,1=3x\\hfill & \\hfill \\\\ \\frac{1}{3}=x\\hfill & \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill & x\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2905820\">To find the <em>y-<\/em>intercept, set[latex]\\,x=0.[\/latex]<\/p>\n<div id=\"fs-id1798574\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=3x-1\\hfill \\\\ y=3\\left(0\\right)-1\\hfill \\\\ y=-1\\hfill \\\\ \\left(0,-1\\right)\\phantom{\\rule{3em}{0ex}}y\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1730534\">We can confirm that our results make sense by observing a graph of the equation as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_012\">(Figure)<\/a>. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<div id=\"Figure_02_01_012\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132956\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1780189\" class=\"textbox key-takeaways\">\n<h3>Given an equation, find the intercepts.<\/h3>\n<ol id=\"fs-id2294904\" type=\"1\">\n<li>Find the <em>x<\/em>-intercept by setting[latex]\\,y=0\\,[\/latex]and solving for[latex]\\,x.[\/latex]<\/li>\n<li>Find the <em>y-<\/em>intercept by setting[latex]\\,x=0\\,[\/latex]and solving for[latex]\\,y.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1689084\">\n<div id=\"fs-id1419163\">\n<h3>Finding the Intercepts of the Given Equation<\/h3>\n<p>Find the intercepts of the equation[latex]\\,y=-3x-4.\\,[\/latex]Then sketch the graph using only the intercepts.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1514793\">Set[latex]\\,y=0\\,[\/latex]to find the <em>x-<\/em>intercept.<\/p>\n<div id=\"fs-id2293732\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\phantom{\\rule{1em}{0ex}}y=-3x-4\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}0=-3x-4\\hfill \\\\ \\phantom{\\rule{1em}{0ex}}4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)\\phantom{\\rule{3em}{0ex}}x\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1560533\">Set[latex]\\,x=0\\,[\/latex]to find the <em>y-<\/em>intercept.<\/p>\n<div id=\"fs-id2803056\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=-3x-4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)\\phantom{\\rule{3.5em}{0ex}}y\\text{\u2212intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points, and draw a line passing through them as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_013\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133002\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id2504052\"><span id=\"fs-id2433829\"><\/details>\n<p><\/span><span id=\"fs-id2433829\"><\/span><span id=\"fs-id2433829\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1518804\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_02\">\n<div id=\"fs-id2437906\">\n<p id=\"fs-id1972959\">Find the intercepts of the equation and sketch the graph:[latex]\\,y=-\\frac{3}{4}x+3.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1447496\"><em>x<\/em>-intercept is[latex]\\,\\left(4,0\\right);[\/latex]<em>y-<\/em>intercept is[latex]\\,\\left(0,3\\right).[\/latex]<\/p>\n<p><span id=\"fs-id1297236\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133035\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1280821\" class=\"bc-section section\">\n<h3>Using the Distance Formula<\/h3>\n<p id=\"fs-id1277804\">Derived from the <span class=\"no-emphasis\">Pythagorean Theorem<\/span>, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,[latex]\\,{a}^{2}+{b}^{2}={c}^{2},[\/latex]is based on a right triangle where <em>a <\/em>and <em>b<\/em> are the lengths of the legs adjacent to the right angle, and <em>c<\/em> is the length of the hypotenuse. See <a class=\"autogenerated-content\" href=\"#Figure_02_01_015\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_02_01_015\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133041\/CNX_CAT_Figure_02_01_015.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x and y axes range from 0 to 7. The points (x sub 1, y sub 1); (x sub 2, y sub 1); and (x sub 2, y sub 2) are labeled and connected to form a triangle. Along the base of the triangle, the following equation is displayed: the absolute value of x sub 2 minus x sub 1 equals a. The hypotenuse of the triangle is labeled: d = c. The remaining side is labeled: the absolute value of y sub 2 minus y sub 1 equals b.\" width=\"487\" height=\"331\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1151919\">The relationship of sides[latex]\\,|{x}_{2}-{x}_{1}|\\,[\/latex]and[latex]\\,|{y}_{2}-{y}_{1}|\\,[\/latex]to side <em>d<\/em> is the same as that of sides <em>a <\/em>and <em>b <\/em>to side <em>c.<\/em> We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example,[latex]\\,|-3|=3.\\,[\/latex]) The symbols[latex]\\,|{x}_{2}-{x}_{1}|\\,[\/latex]and[latex]\\,|{y}_{2}-{y}_{1}|\\,[\/latex]indicate that the lengths of the sides of the triangle are positive. To find the length <em>c<\/em>, take the square root of both sides of the Pythagorean Theorem.<\/p>\n<div id=\"fs-id2730208\" class=\"unnumbered aligncenter\">[latex]{c}^{2}={a}^{2}+{b}^{2}\\to c=\\sqrt{{a}^{2}+{b}^{2}}[\/latex]<\/div>\n<p id=\"fs-id2666328\">It follows that the distance formula is given as<\/p>\n<div id=\"fs-id1188836\" class=\"unnumbered aligncenter\">[latex]{d}^{2}={\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}\\to d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<p id=\"fs-id1832560\">We do not have to use the absolute value symbols in this definition because any number squared is positive.<\/p>\n<div id=\"fs-id1521890\" class=\"textbox key-takeaways\">\n<h3>The Distance Formula<\/h3>\n<p id=\"eip-id2982180\">Given endpoints[latex]\\,\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and[latex]\\,\\left({x}_{2},{y}_{2}\\right),[\/latex]the distance between two points is given by<\/p>\n<div id=\"fs-id2933949\" class=\"unnumbered aligncenter\">[latex]d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_02_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1203380\">\n<div id=\"fs-id1204935\">\n<h3>Finding the Distance between Two Points<\/h3>\n<p id=\"fs-id1418038\">Find the distance between the points[latex]\\,\\left(-3,-1\\right)\\,[\/latex]and[latex]\\,\\left(2,3\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3223093\">Let us first look at the graph of the two points. Connect the points to form a right triangle as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_016\">(Figure)<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_01_016\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133043\/CNX_CAT_Figure_02_01_016.jpg\" alt=\"This is an image of a triangle on an x, y coordinate plane. The x-axis ranges from negative 4 to 4. The y-axis ranges from negative 2 to 4. The points (-3, -1); (2, -1); and (2, 3) are plotted and labeled on the graph. The points are connected to form a triangle\" width=\"487\" height=\"289\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1211736\">Then, calculate the length of <em>d <\/em>using the distance formula.<\/p>\n<div id=\"fs-id1839331\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}}\\hfill \\\\ d=\\sqrt{{\\left(2-\\left(-3\\right)\\right)}^{2}+{\\left(3-\\left(-1\\right)\\right)}^{2}}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{{\\left(5\\right)}^{2}+{\\left(4\\right)}^{2}}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{25+16}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{41}\\hfill \\end{array}\\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1574404\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_03\">\n<div id=\"fs-id1938614\">\n<p id=\"fs-id1938615\">Find the distance between two points:[latex]\\,\\left(1,4\\right)\\,[\/latex]and[latex]\\,\\left(11,9\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2667607\">[latex]\\sqrt{125}=5\\sqrt{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_01_06\" class=\"textbox examples\">\n<div id=\"fs-id2503622\">\n<div id=\"fs-id1338610\">\n<h3>Finding the Distance between Two Locations<\/h3>\n<p id=\"fs-id2522450\">Let\u2019s return to the situation introduced at the beginning of this section.<\/p>\n<p id=\"fs-id2640218\">Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in <a class=\"autogenerated-content\" href=\"#Figure_02_01_001\">(Figure)<\/a>. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2522863\">The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at[latex]\\,\\left(1,1\\right).\\,[\/latex]The next stop is 5 blocks to the east, so it is at[latex]\\,\\left(5,1\\right).\\,[\/latex]After that, she traveled 3 blocks east and 2 blocks north to[latex]\\,\\left(8,3\\right).\\,[\/latex]Lastly, she traveled 4 blocks north to[latex]\\,\\left(8,7\\right).\\,[\/latex]We can label these points on the grid as in <a class=\"autogenerated-content\" href=\"#Figure_02_01_017\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_02_01_017\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133101\/CNX_CAT_Figure_02_01_017.jpg\" alt=\"This is an image of a road map of a city. The point (1, 1) is on North Avenue and Bertau Avenue. The point (5, 1) is on North Avenue and Wolf Road. The point (8, 3) is on Mannheim Road and McLean Street. The point (8, 7) is on Mannheim Road and Schiller Avenue.\" width=\"731\" height=\"480\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id2777115\">Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.<\/p>\n<ul id=\"fs-id1688170\">\n<li>From her starting location to her first stop at[latex]\\,\\left(1,1\\right),[\/latex]Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.<\/li>\n<li>Her second stop is at[latex]\\,\\left(5,1\\right).\\,[\/latex]So from[latex]\\,\\left(1,1\\right)\\,[\/latex]to[latex]\\,\\left(5,1\\right),[\/latex]Tracie drove east 4,000 feet.<\/li>\n<li>Her third stop is at[latex]\\,\\left(8,3\\right).\\,[\/latex]There are a number of routes from[latex]\\,\\left(5,1\\right)\\,[\/latex]to[latex]\\,\\left(8,3\\right).\\,[\/latex]Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let\u2019s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.<\/li>\n<li>Tracie\u2019s final stop is at[latex]\\,\\left(8,7\\right).\\,[\/latex]This is a straight drive north from[latex]\\,\\left(8,3\\right)\\,[\/latex]for a total of 4,000 feet.<\/li>\n<\/ul>\n<p id=\"fs-id1688205\">Next, we will add the distances listed in <a class=\"autogenerated-content\" href=\"#Table_02_01_03\">(Figure)<\/a>.<\/p>\n<table id=\"Table_02_01_03\" summary=\"A table with 6 rows and 2 columns. The entries in the first row are: From\/To and Number of Feet Driven. The entries in the second row are: (0, 0) to (1, 1) and 2,000. The entries in the third row are: (1, 1) to (5, 1) and 4,000. The entries in the fourth row are: (5, 1) to (8, 3) and 5,000. The entries in the fourth row are: (8, 3) to (8, 7) and 4,000. The entries in the sixth row are: Total and 15,000.\">\n<thead>\n<tr>\n<th>From\/To<\/th>\n<th>Number of Feet Driven<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(0,0\\right)\\,[\/latex]to[latex]\\,\\left(1,1\\right)[\/latex]<\/td>\n<td>2,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(1,1\\right)\\,[\/latex]to[latex]\\left(5,1\\right)\\,[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,1\\right)\\,[\/latex]to[latex]\\,\\left(8,3\\right)[\/latex]<\/td>\n<td>5,000<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(8,3\\right)\\,[\/latex]to[latex]\\,\\left(8,7\\right)[\/latex]<\/td>\n<td>4,000<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>15,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id2629595\">The total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points[latex]\\,\\left(0,0\\right)\\,[\/latex]and[latex]\\,\\left(8,7\\right).[\/latex]<\/p>\n<div id=\"fs-id2959388\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}d=\\sqrt{{\\left(8-0\\right)}^{2}+{\\left(7-0\\right)}^{2}}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{64+49}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=\\sqrt{113}\\hfill \\\\ \\phantom{\\rule{.7em}{0ex}}=10.63\\text{ units}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1475568\">At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point[latex]\\,\\left(8,7\\right).\\,[\/latex]Perhaps you have heard the saying \u201cas the crow flies,\u201d which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2507035\" class=\"bc-section section\">\n<h3>Using the Midpoint Formula<\/h3>\n<p id=\"fs-id1151538\">When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment,[latex]\\,\\left({x}_{1},{y}_{1}\\right)\\,[\/latex]and[latex]\\,\\left({x}_{2},{y}_{2}\\right),[\/latex]the midpoint formula states how to find the coordinates of the midpoint[latex]\\,M.[\/latex]<\/p>\n<div id=\"fs-id1520430\" class=\"unnumbered aligncenter\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\n<p id=\"fs-id2837053\">A graphical view of a midpoint is shown in <a class=\"autogenerated-content\" href=\"#Figure_02_01_018\">(Figure)<\/a>. Notice that the line segments on either side of the midpoint are congruent.<\/p>\n<div id=\"Figure_02_01_018\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133109\/CNX_CAT_Figure_02_01_018.jpg\" alt=\"This is a line graph on an x, y coordinate plane with the x and y axes ranging from 0 to 6. The points (x sub 1, y sub 1), (x sub 2, y sub 2), and (x sub 1 plus x sub 2 all over 2, y sub 1 plus y sub 2 all over 2) are plotted. A straight line runs through these three points. Pairs of short parallel lines bisect the two sections of the line to note that they are equivalent.\" width=\"487\" height=\"290\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 15.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"Example_02_01_07\" class=\"textbox examples\">\n<div id=\"fs-id3008576\">\n<div id=\"fs-id3008579\">\n<h3>Finding the Midpoint of the Line Segment<\/h3>\n<p id=\"fs-id1336692\">Find the midpoint of the line segment with the endpoints[latex]\\,\\left(7,-2\\right)\\,[\/latex]and[latex]\\,\\left(9,5\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1213113\">Use the formula to find the midpoint of the line segment.<\/p>\n<div id=\"fs-id1926574\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\phantom{\\rule{6.5em}{0ex}}=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1537533\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_01_04\">\n<div id=\"fs-id2434160\">\n<p id=\"fs-id1824194\">Find the midpoint of the line segment with endpoints[latex]\\,\\left(-2,-1\\right)\\,[\/latex]and[latex]\\,\\left(-8,6\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3008559\">[latex]\\left(-5,\\frac{5}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_01_08\" class=\"textbox examples\">\n<div id=\"fs-id1977814\">\n<div id=\"fs-id1977817\">\n<h3>Finding the Center of a Circle<\/h3>\n<p id=\"fs-id1977163\">The diameter of a circle has endpoints[latex]\\,\\left(-1,-4\\right)\\,[\/latex]and[latex]\\,\\left(5,-4\\right).\\,[\/latex]Find the center of the circle.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1523294\">The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.<\/p>\n<div id=\"fs-id2858239\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4-4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1539521\" class=\"precalculus media\">\n<p id=\"fs-id2435248\">Access these online resources for additional instruction and practice with the Cartesian coordinate system.<\/p>\n<ul id=\"fs-id2766408\">\n<li><a href=\"http:\/\/Openstaxcollege.org\/l\/coordplotpnts\">Plotting points on the coordinate plane<\/a><\/li>\n<li><a href=\"http:\/\/Openstaxcollege.org\/l\/xyintsgraph\">Find x and y intercepts based on the graph of a line<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id2721419\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1449332\">\n<li>We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the <em>x-<\/em>axis and displacement from the <em>y-<\/em>axis. See <a class=\"autogenerated-content\" href=\"#Example_02_01_01\">(Figure)<\/a>.<\/li>\n<li>An equation can be graphed in the plane by creating a table of values and plotting points. See <a class=\"autogenerated-content\" href=\"#Example_02_01_02\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form <em>y=<\/em>_____. See <a class=\"autogenerated-content\" href=\"#Example_02_01_03\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>Finding the <em>x- <\/em>and <em>y-<\/em>intercepts can define the graph of a line. These are the points where the graph crosses the axes. See <a class=\"autogenerated-content\" href=\"#Example_02_01_04\">(Figure)<\/a>.<\/li>\n<li>The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See <a class=\"autogenerated-content\" href=\"#Example_02_01_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_01_06\">(Figure)<\/a>.<\/li>\n<li>The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the <em>x<\/em>-coordinates and the sum of the <em>y<\/em>-coordinates of the endpoints by 2. See <a class=\"autogenerated-content\" href=\"#Example_02_01_07\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_01_08\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1553580\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id2496131\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1355431\">\n<div id=\"fs-id1355432\">\n<p id=\"fs-id1355433\">Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1543668\">Answers may vary. Yes. It is possible for a point to be on the <em>x<\/em>-axis or on the <em>y<\/em>-axis and therefore is considered to NOT be in one of the quadrants.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2435397\">\n<div>\n<p id=\"fs-id1929270\">Describe the process for finding the <em>x-<\/em>intercept and the <em>y<\/em>-intercept of a graph algebraically.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1351774\">\n<div id=\"fs-id2682309\">\n<p id=\"fs-id2682310\">Describe in your own words what the <em>y<\/em>-intercept of a graph is.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1322043\">The <em>y<\/em>-intercept is the point where the graph crosses the <em>y<\/em>-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1823207\">\n<div id=\"fs-id1823208\">\n<p id=\"fs-id1823209\">When using the distance formula[latex]\\,d=\\sqrt{{\\left({x}_{2}-{x}_{1}\\right)}^{2}+{\\left({y}_{2}-{y}_{1}\\right)}^{2}},[\/latex]explain the correct order of operations that are to be performed to obtain the correct answer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3263952\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1789811\">For each of the following exercises, find the <em>x<\/em>-intercept and the <em>y<\/em>-intercept without graphing. Write the coordinates of each intercept.<\/p>\n<div id=\"fs-id3039754\">\n<div id=\"fs-id3039755\">\n<p id=\"fs-id2512845\">[latex]y=-3x+6[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1441309\">The <em>x-<\/em>intercept is[latex]\\,\\left(2,0\\right)\\,[\/latex]and the <em>y<\/em>-intercept is[latex]\\,\\left(0,6\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1499169\">\n<div id=\"fs-id1499170\">\n<p id=\"fs-id1499171\">[latex]4y=2x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1007678\">\n<div id=\"fs-id1591152\">\n<p id=\"fs-id1591153\">[latex]3x-2y=6[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1769437\">The <em>x-<\/em>intercept is[latex]\\,\\left(2,0\\right)\\,[\/latex]and the <em>y<\/em>-intercept is[latex]\\,\\left(0,-3\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2948006\">\n<div id=\"fs-id2948007\">\n<p id=\"fs-id2948008\">[latex]4x-3=2y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1723142\">\n<div id=\"fs-id1723143\">\n<p id=\"fs-id1937401\">[latex]3x+8y=9[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2737079\">The <em>x-<\/em>intercept is[latex]\\,\\left(3,0\\right)\\,[\/latex]and the <em>y<\/em>-intercept is[latex]\\,\\left(0,\\frac{9}{8}\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1197890\">\n<div id=\"fs-id1197891\">\n<p id=\"fs-id1197892\">[latex]2x-\\frac{2}{3}=\\frac{3}{4}y+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id3207567\">For each of the following exercises, solve the equation for <em>y<\/em> in terms of <em>x<\/em>.<\/p>\n<div id=\"fs-id3042175\">\n<div id=\"fs-id3042176\">\n<p id=\"fs-id2486646\">[latex]4x+2y=8[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1830748\">[latex]y=4-2x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1238095\">\n<div id=\"fs-id1238096\">\n<p id=\"fs-id1238097\">[latex]3x-2y=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2512516\">\n<div id=\"fs-id1441357\">\n<p id=\"fs-id1441358\">[latex]2x=5-3y[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1939378\">[latex]y=\\frac{5-2x}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1467950\">\n<div id=\"fs-id2521131\">\n<p id=\"fs-id2521132\">[latex]x-2y=7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1278706\">\n<div id=\"fs-id1402803\">\n<p id=\"fs-id1402804\">[latex]5y+4=10x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1688278\">[latex]y=2x-\\frac{4}{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2388766\">\n<div id=\"fs-id2388767\">\n<p id=\"fs-id2952969\">[latex]5x+2y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1702384\">For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.<\/p>\n<div id=\"fs-id1798648\">\n<div id=\"fs-id1798650\">\n<p id=\"fs-id1402574\">[latex]\\left(-4,1\\right)\\,[\/latex]and[latex]\\,\\left(3,-4\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1520540\">[latex]d=\\sqrt{74}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1762329\">\n<div id=\"fs-id2947043\">\n<p id=\"fs-id2947044\">[latex]\\left(2,-5\\right)\\,[\/latex]and[latex]\\,\\left(7,4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1932412\">\n<div id=\"fs-id2266178\">\n<p id=\"fs-id2266179\">[latex]\\left(5,0\\right)\\,[\/latex]and[latex]\\,\\left(5,6\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2729922\">[latex]d=\\sqrt{36}=6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1538085\">\n<div id=\"fs-id1538086\">\n<p id=\"fs-id1538087\">[latex]\\left(-4,3\\right)\\,[\/latex]and[latex]\\,\\left(10,3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2628482\">\n<div id=\"fs-id2628483\">\n<p id=\"fs-id2628484\">Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.<\/p>\n<p id=\"fs-id2644287\">[latex]\\left(19,12\\right)\\,[\/latex]and[latex]\\,\\left(41,71\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2411257\">[latex]d\\approx 62.97[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2753833\">For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.<\/p>\n<div id=\"fs-id2389564\">\n<div id=\"fs-id2389565\">\n<p id=\"fs-id2389566\">[latex]\\left(-5,-6\\right)\\,[\/latex]and[latex]\\,\\left(4,2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1386855\">\n<div id=\"fs-id1386856\">\n<p id=\"fs-id1386857\">[latex]\\left(-1,1\\right)\\,[\/latex]and[latex]\\,\\left(7,-4\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1225500\">[latex]\\left(3,\\frac{-3}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2425333\">\n<div id=\"fs-id2425334\">\n<p id=\"fs-id2425335\">[latex]\\left(-5,-3\\right)\\,[\/latex]and[latex]\\,\\left(-2,-8\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2736528\">\n<div id=\"fs-id2736529\">\n<p id=\"fs-id2736530\">[latex]\\left(0,7\\right)\\,[\/latex]and[latex]\\,\\left(4,-9\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2694213\">[latex]\\left(2,-1\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1960033\">\n<div id=\"fs-id1960034\">\n<p id=\"fs-id2431256\">[latex]\\left(-43,17\\right)\\,[\/latex]and[latex]\\,\\left(23,-34\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2434980\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1919734\">For each of the following exercises, identify the information requested.<\/p>\n<div id=\"fs-id1940524\">\n<div id=\"fs-id1940525\">\n<p id=\"fs-id1422420\">What are the coordinates of the origin?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3162327\">[latex]\\left(0,0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1965313\">\n<div id=\"fs-id1965314\">\n<p id=\"fs-id1965315\">If a point is located on the <em>y<\/em>-axis, what is the <em>x<\/em>-coordinate?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1575040\">\n<div id=\"fs-id1575041\">\n<p id=\"fs-id1575042\">If a point is located on the <em>x<\/em>-axis, what is the <em>y<\/em>-coordinate?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1583783\">[latex]y=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1475798\">For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).<\/p>\n<div id=\"fs-id1182006\">\n<div id=\"fs-id1182007\">\n<p id=\"fs-id1182008\">[latex]\\left(4,1\\right)\\left(-2,-3\\right)\\left(5,0\\right)[\/latex]<\/p>\n<p><span id=\"fs-id2387110\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133120\/CNX_CAT_Figure_02_01_201.jpg\" alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1341660\">\n<div id=\"fs-id1341661\">\n<p id=\"fs-id1341662\">[latex]\\left(-1,2\\right)\\left(0,4\\right)\\left(2,1\\right)[\/latex]<\/p>\n<p><span id=\"fs-id3165041\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133120\/CNX_CAT_Figure_02_01_201.jpg\" alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133124\/CNX_CAT_Figure_02_01_203.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (0,4); (-1,2) and (2,1) are plotted and labeled.\" \/>not collinear<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2440944\">\n<div id=\"fs-id2440946\">\n<p id=\"fs-id2440947\">[latex]\\left(-3,0\\right)\\left(-3,4\\right)\\left(-3,-3\\right)[\/latex]<\/p>\n<p><span id=\"fs-id2918881\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133120\/CNX_CAT_Figure_02_01_201.jpg\" alt=\"This is an image of a blank x, y coordinate plane with the x and y axes ranging from negative 5 to 5.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2453414\">\n<div id=\"fs-id2453415\">\n<p id=\"fs-id2453416\">Name the coordinates of the points graphed.<\/p>\n<p><span id=\"fs-id1333042\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133131\/CNX_CAT_Figure_02_01_205.jpg\" alt=\"This is an image of an x, y coordinate plane where the x and y-axis range from negative 5 to 5. Three points are plotted: A, B, and C.\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3158703\">[latex]\\left(-3,2\\right),\\left(1,3\\right),\\left(4,0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3155305\">\n<div id=\"fs-id3155306\">\n<p id=\"fs-id3155307\">Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.<\/p>\n<p id=\"fs-id1844064\">[latex]\\begin{array}{l}a.\\left(-3,-4\\right)\\\\ b.\\left(-5,0\\right)\\\\ c.\\left(1,-4\\right)\\\\ d.\\left(-2,7\\right)\\\\ e.\\left(0,-3\\right)\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1569563\">For each of the following exercises, construct a table and graph the equation by plotting at least three points.<\/p>\n<div id=\"fs-id1569567\">\n<div id=\"fs-id3182657\">\n<p id=\"fs-id3182658\">[latex]y=\\frac{1}{3}x+2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id2801280\" summary=\"A table with 5 rows and 2 columns. The entries in the first row are: x and y. The entries in the second row are: negative 3 and 1. The entries in the third row are: 0 and 2. The entries in the fourth row are: 3 and 3. The entries in the fifth row are: 6 and 4.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span id=\"fs-id1920364\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133143\/CNX_CAT_Figure_02_01_206.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (-3, 1); (0, 2); (3, 3) and (6, 4) are plotted and labeled. A line runs through all these points.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3070032\">\n<div id=\"fs-id3070034\">\n<p id=\"fs-id3070035\">[latex]y=-3x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1445099\">\n<div id=\"fs-id1477477\">\n<p id=\"fs-id1477478\">[latex]2y=x+3[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id1764494\" summary=\"A table with 4 rows and 2 columns. The entries in the first row are: x and y. The entries in the second row are: negative 3 and 0. The entries in the third row are: 0 and 1.5. The entries in the fourth row are: 3 and 3.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><em>x<\/em><\/td>\n<td><em>y<\/em><\/td>\n<\/tr>\n<tr>\n<td>\u20133<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>1.5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span id=\"fs-id1269392\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133151\/CNX_CAT_Figure_02_01_208.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (-3, 0); (0, 1.5) and (3, 3) are plotted and labeled. A line runs through all of these points.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1387762\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id2528941\">For each of the following exercises, find and plot the <em>x-<\/em> and <em>y<\/em>-intercepts, and graph the straight line based on those two points.<\/p>\n<div id=\"fs-id766182\">\n<div id=\"fs-id766183\">\n<p id=\"fs-id766184\">[latex]4x-3y=12[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3176745\">\n<div id=\"fs-id3176746\">\n<p id=\"fs-id3176748\">[latex]x-2y=8[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1333412\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133158\/CNX_CAT_Figure_02_01_210.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (8, 0) and (0, -4) are plotted and labeled. A line runs through both of these points.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1272882\">\n<div id=\"fs-id1272883\">\n<p id=\"fs-id1272884\">[latex]y-5=5x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1951937\">\n<div id=\"fs-id1951938\">\n<p id=\"fs-id1951940\">[latex]3y=-2x+6[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2385560\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133207\/CNX_CAT_Figure_02_01_212.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 10 to 10. The points (0, 2) and (3, 0) are plotted and labeled. A line runs through both of these points.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1517685\">\n<div id=\"fs-id1929175\">\n<p id=\"fs-id1929176\">[latex]y=\\frac{x-3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1919624\">For each of the following exercises, use the graph in the figure below.<\/p>\n<p><span id=\"fs-id1832449\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133214\/CNX_CAT_Figure_02_01_214.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 5 to 5. The points (-3, 4) and (5, 2) are plotted. A line connects these two points.\" \/><\/span><\/p>\n<div id=\"fs-id1267899\">\n<div id=\"fs-id2432278\">\n<p id=\"fs-id2432279\">Find the distance between the two endpoints using the distance formula. Round to three decimal places.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2905920\">[latex]d=8.246[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1895437\">\n<div id=\"fs-id1815372\">\n<p id=\"fs-id1815373\">Find the coordinates of the midpoint of the line segment connecting the two points.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1815376\">\n<div id=\"fs-id1418769\">\n<p id=\"fs-id1418770\">Find the distance that[latex]\\,\\left(-3,4\\right)\\,[\/latex]is from the origin.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1258397\">[latex]d=5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1892569\">\n<div id=\"fs-id1892570\">\n<p id=\"fs-id1892571\">Find the distance that[latex]\\,\\left(5,2\\right)\\,[\/latex]is from the origin. Round to three decimal places.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2437516\">\n<div id=\"fs-id2437517\">\n<p id=\"fs-id1691144\">Which point is closer to the origin?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1691149\">[latex]\\left(-3,4\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2437604\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1333818\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\n<p id=\"fs-id1333822\">After graphing it, use the 2<sup>nd<\/sup> CALC button and 1:value button, hit enter. At the lower part of the screen you will see \u201cx=\u201d and a blinking cursor. You may enter any number for <em>x<\/em> and it will display the <em>y<\/em> value for any <em>x<\/em> value you input. Use this and plug in <em>x<\/em> = 0, thus finding the <em>y<\/em>-intercept, for each of the following graphs.<\/p>\n<div id=\"fs-id1686604\">\n<div id=\"fs-id1686605\">\n<p id=\"fs-id1686606\">[latex]{\\text{Y}}_{1}=-2x+5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1181943\">\n<div id=\"fs-id1181944\">\n<p id=\"fs-id1181945\">[latex]{\\text{Y}}_{1}=\\frac{3x-8}{4}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2639345\">[latex]x=0\\text{ }y=-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2800083\">\n<div id=\"fs-id1421792\">\n<p id=\"fs-id1421793\">[latex]{\\text{Y}}_{1}=\\frac{x+5}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2785067\">For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.<\/p>\n<p id=\"fs-id2384791\">After graphing it, use the 2<sup>nd<\/sup> CALC button and 2:zero button, hit enter. At the lower part of the screen you will see \u201cleft bound?\u201d and a blinking cursor on the graph of the line. Move this cursor to the left of the <em>x<\/em>-intercept, hit ENTER. Now it says \u201cright bound?\u201d Move the cursor to the right of the <em>x<\/em>-intercept, hit enter. Now it says \u201cguess?\u201d Move your cursor to the left somewhere in between the left and right bound near the <em>x<\/em>-intercept. Hit enter. At the bottom of your screen it will display the coordinates of the <em>x-<\/em>intercept or the \u201czero\u201d to the <em>y<\/em>-value. Use this to find the <em>x<\/em>-intercept.<\/p>\n<p id=\"fs-id2016100\">Note: With linear\/straight line functions the zero is not really a \u201cguess,\u201d but it is necessary to enter a \u201cguess\u201d so it will search and find the exact <em>x<\/em>-intercept between your right and left boundaries. With other types of functions (more than one <em>x<\/em>-intercept), they may be irrational numbers so \u201cguess\u201d is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.<\/p>\n<div id=\"fs-id1425403\">\n<div id=\"fs-id1425404\">\n<p id=\"fs-id1425405\">[latex]{\\text{Y}}_{1}=-8x+6[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2016707\">[latex]x=0.75\\text{ }y=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1845247\">\n<div id=\"fs-id1845248\">\n<p id=\"fs-id1845249\">[latex]{\\text{Y}}_{1}=4x-7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2387375\">\n<div id=\"fs-id2387376\">\n<p id=\"fs-id1673534\">[latex]{\\text{Y}}_{1}=\\frac{3x+5}{4}\\,[\/latex]Round your answer to the nearest thousandth.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2781155\">[latex]x=-1.667\\text{ }y=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1932603\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1932608\">\n<div id=\"fs-id1333979\">\n<p id=\"fs-id1333980\">A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2513497\">\n<div id=\"fs-id2513498\">\n<p id=\"fs-id2513499\">If the road was made in the previous exercise, how much shorter would the man\u2019s one-way trip be every day?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1513803\">[latex]\\text{15}\\text{\u221211}.\\text{2 }=\\text{ 3}.8\\,[\/latex]mi shorter<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2508338\">\n<div id=\"fs-id2508339\">\n<p id=\"fs-id1803338\">Given these four points:[latex]\\,A\\left(1,3\\right),\\text{}B\\left(-3,5\\right),\\text{}C\\left(4,7\\right),\\text{ and }D\\left(5,-4\\right),[\/latex]find the coordinates of the midpoint of line segments[latex]\\,\\overline{\\text{AB}}\\,[\/latex]and[latex]\\,\\overline{\\text{CD}}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1343510\">\n<div id=\"fs-id1343511\">\n<p id=\"fs-id1343512\">After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1960062\">[latex]\\text{6}.0\\text{42}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1312332\">\n<div id=\"fs-id1312333\">\n<p id=\"fs-id3207842\">Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.<\/p>\n<p><span id=\"fs-id3207847\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133216\/CNX_CAT_Figure_02_01_215.jpg\" alt=\"This is an image of an x, y coordinate plane with the x and y axes ranging from negative 12 to 12. The points (-6, 5); (10, 5); (-6, -1) and (10, -1) are plotted and labeled. These points are connected to form a rectangle. Dotted lines extend from each corner point to their opposite point.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1228174\">\n<div id=\"fs-id1228176\">\n<p id=\"fs-id1228177\">In the previous exercise, find the coordinates of the midpoint for each diagonal.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1542004\">Midpoint of each diagonal is the same point[latex]\\,\\left(2,2\\right).\\,[\/latex]Note this is a characteristic of rectangles, but not other quadrilaterals.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2454474\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id2722618\">\n<div id=\"fs-id2722619\">\n<p id=\"fs-id2722620\">The coordinates on a map for San Francisco are[latex]\\,\\left(53,17\\right)\\,[\/latex]and those for Sacramento are[latex]\\,\\left(123,78\\right).\\,[\/latex]Note that coordinates represent miles. Find the distance between the cities to the nearest mile.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1511688\">\n<div id=\"fs-id1511690\">\n<p id=\"fs-id1511691\">If San Jose\u2019s coordinates are[latex]\\,\\left(76,-12\\right),[\/latex]where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2700244\">[latex]\\text{37}\\,[\/latex]mi<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2925632\">\n<div id=\"fs-id2925633\">\n<p id=\"fs-id2925634\">A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were[latex]\\,\\left(49,64\\right).\\,[\/latex]One rescue boat is at the coordinates[latex]\\,\\left(60,82\\right)\\,[\/latex]and a second Coast Guard craft is at coordinates[latex]\\,\\left(58,47\\right).\\,[\/latex]Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1551836\">\n<div id=\"fs-id1551837\">\n<p id=\"fs-id1551838\">A man on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?<\/p>\n<p><span id=\"fs-id1477367\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133228\/CNX_CAT_Figure_02_01_217.jpg\" alt=\"A right triangle with its bottom left point sitting on the point (0,0). The upper right hand corner is labeled (20,50). The base has a length of 20 units and the triangle has a height of 50 units.\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1783293\">54 ft<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1717974\">\n<div id=\"fs-id1717976\">\n<p id=\"fs-id1717977\">If we rent a truck and pay a $75\/day fee plus $.20 for every mile we travel, write a linear equation that would express the total cost[latex]\\,y,[\/latex]using[latex]\\,x\\,[\/latex]to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1890952\">\n<dt>Cartesian coordinate system<\/dt>\n<dd id=\"fs-id1402320\">a grid system designed with perpendicular axes invented by Ren\u00e9 Descartes<\/dd>\n<\/dl>\n<dl id=\"fs-id1402323\">\n<dt>distance formula<\/dt>\n<dd id=\"fs-id1574693\">a formula that can be used to find the length of a line segment if the endpoints are known<\/dd>\n<\/dl>\n<dl id=\"fs-id1574696\">\n<dt>equation in two variables<\/dt>\n<dd id=\"fs-id1277856\">a mathematical statement, typically written in <em>x <\/em>and <em>y<\/em>, in which two expressions are equal<\/dd>\n<\/dl>\n<dl id=\"fs-id2508809\">\n<dt>graph in two variables<\/dt>\n<dd id=\"fs-id2508812\">the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1277986\">\n<dt>intercepts<\/dt>\n<dd id=\"fs-id1277990\">the points at which the graph of an equation crosses the <em>x<\/em>-axis and the <em>y<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1337623\">\n<dt>midpoint formula<\/dt>\n<dd id=\"fs-id3082583\">a formula to find the point that divides a line segment into two parts of equal length<\/dd>\n<\/dl>\n<dl id=\"fs-id3082586\">\n<dt>ordered pair<\/dt>\n<dd id=\"fs-id1223328\">a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, [latex]\\,\\left(x,y\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id2502726\">\n<dt>origin<\/dt>\n<dd id=\"fs-id1534441\">the point where the two axes cross in the center of the plane, described by the ordered pair[latex]\\,\\left(0,0\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id3067555\">\n<dt>quadrant<\/dt>\n<dd id=\"fs-id2875924\">one quarter of the coordinate plane, created when the axes divide the plane into four sections<\/dd>\n<\/dl>\n<dl id=\"fs-id2875927\">\n<dt><em>x<\/em>-axis<\/dt>\n<dd id=\"fs-id2389105\">the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right<\/dd>\n<\/dl>\n<dl id=\"fs-id1504116\">\n<dt><em>x-<\/em>coordinate<\/dt>\n<dd id=\"fs-id2302378\">the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1227889\">\n<dt><em>x-<\/em>intercept<\/dt>\n<dd id=\"fs-id1278364\">the point where a graph intersects the <em>x-<\/em>axis; an ordered pair with a <em>y<\/em>-coordinate of zero<\/dd>\n<\/dl>\n<dl id=\"fs-id1257845\">\n<dt><em>y<\/em>-axis<\/dt>\n<dd id=\"fs-id815935\">the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top<\/dd>\n<\/dl>\n<dl id=\"fs-id2629843\">\n<dt><em>y-<\/em>coordinate<\/dt>\n<dd id=\"fs-id1521234\">the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id2364783\">\n<dt><em>y<\/em>-intercept<\/dt>\n<dd id=\"fs-id1575741\">a point where a graph intercepts the <em>y-<\/em>axis; an ordered pair with an <em>x<\/em>-coordinate of zero<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n","protected":false},"author":291,"menu_order":2,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-36","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/36","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\/36\/revisions"}],"predecessor-version":[{"id":37,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/36\/revisions\/37"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/36\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=36"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=36"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=36"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=36"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}