{"id":1976,"date":"2021-08-19T16:03:43","date_gmt":"2021-08-19T16:03:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-area-between-two-curves\/"},"modified":"2021-11-17T01:52:28","modified_gmt":"2021-11-17T01:52:28","slug":"skills-review-for-area-between-two-curves","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-area-between-two-curves\/","title":{"raw":"Skills Review for Area Between Two Curves, Determining Volumes by Slicing, and Volumes of Revolution: Cylindrical Shells","rendered":"Skills Review for Area Between Two Curves, Determining Volumes by Slicing, and Volumes of Revolution: Cylindrical Shells"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph horizontal and vertical lines<\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Graph linear equations in different forms using ordered pairs&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4610,&quot;4&quot;:[null,2,13624051],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Graph linear equations in different forms using ordered pairs<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Graph a linear equation using x and y-intercepts&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4610,&quot;4&quot;:[null,2,13624051],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Graph a linear equation using x and y-intercepts<\/span><\/li>\r\n \t<li>Determine if and where two equations intersect<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the first three sections of Module 6, it is very important that we graph the equations we are given so there is a visual to assist us with setting up the needed integral. First, we will review the special equation forms for horizontal and vertical lines. Then, we will review how to graph equations using points and intercepts and determine where two functions intersect.\r\n<h2>Graph Vertical and Horizontal Lines<\/h2>\r\nThe equation of a <strong>vertical line<\/strong> is given as\r\n<div style=\"text-align: center;\">[latex]x=c[\/latex]<\/div>\r\nwhere <em>c <\/em>is a constant. The slope of a vertical line is undefined, and regardless of the <em>y-<\/em>value of any point on the line, the <em>x-<\/em>coordinate of the point will be <em>c<\/em>.\r\n\r\nSuppose that we want to find the equation of a line containing the following points: [latex]\\left(-3,-5\\right),\\left(-3,1\\right),\\left(-3,3\\right)[\/latex], and [latex]\\left(-3,5\\right)[\/latex]. First, we will find the slope.\r\n<div style=\"text-align: center;\">[latex]m=\\frac{5 - 3}{-3-\\left(-3\\right)}=\\frac{2}{0}[\/latex]<\/div>\r\nZero in the denominator means that the slope is undefined and, therefore, we cannot use point-slope form. However, we can plot the points. Notice that all of the <em>x-<\/em>coordinates are the same and we find a vertical line through [latex]x=-3[\/latex].\r\n\r\nThe equation of a <strong>horizontal line<\/strong> is given as\r\n<div style=\"text-align: center;\">[latex]y=c[\/latex]<\/div>\r\nwhere <em>c <\/em>is a constant. The slope of a horizontal line is zero, and for any <em>x-<\/em>value of a point on the line, the <em>y-<\/em>coordinate will be <em>c<\/em>.\r\n\r\nSuppose we want to find the equation of a line that contains the following set of points: [latex]\\left(-2,-2\\right),\\left(0,-2\\right),\\left(3,-2\\right)[\/latex], and [latex]\\left(5,-2\\right)[\/latex]. We can use point-slope form. First, we find the slope using any two points on the line.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m=\\frac{-2-\\left(-2\\right)}{0-\\left(-2\\right)}\\hfill \\\\ =\\frac{0}{2}\\hfill \\\\ =0\\hfill \\end{array}[\/latex]<\/div>\r\nUse any point for [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] in the formula, or use the <em>y<\/em>-intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-\\left(-2\\right)=0\\left(x - 3\\right)\\hfill \\\\ y+2=0\\hfill \\\\ y=-2\\hfill \\end{array}[\/latex]<\/div>\r\nThe graph is a horizontal line through [latex]y=-2[\/latex]. Notice that all of the <em>y-<\/em>coordinates are the same.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185925\/CNX_CAT_Figure_02_02_003.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\" width=\"487\" height=\"367\" \/> The line <i>x<\/i> = \u22123 is a vertical line. The line <i>y<\/i> = \u22122 is a horizontal line.[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing Horizontal and Vertical Lines<\/h3>\r\nGraph the line [latex]x=2[\/latex].\r\n[reveal-answer q=\"122244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"122244\"]\r\n\r\nSince [latex]x=[\/latex] is the equation form for a vertical line, we draw a vertical line that goes through 2 on the\u00a0<em>x<\/em>-axis.\r\n\r\n<img class=\"size-medium wp-image-468 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5604\/2021\/05\/20214008\/Vertical-Line_a-300x281.jpg\" alt=\"A vertical line that goes through a value of 2 on the x-axis\" width=\"300\" height=\"281\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]222262[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]222263[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use Points to Graph Equations<\/h2>\r\n<div>\r\n\r\nWe can plot a set of points to represent an equation. 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\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.\r\n<table 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)\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=2x - 1[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n<td>[latex]\\left(-3,-7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n<td>[latex]\\left(-2,-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n<td>[latex]\\left(0,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=2\\left(1\\right)-1=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n<td>[latex]\\left(3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can plot these points from the table. The points for this particular equation form a line, so we can connect them.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/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\" \/>\r\n\r\nNote 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.\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, graph by plotting points<\/h3>\r\n<ol>\r\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>\r\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 graphing easier.<\/li>\r\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>\r\n \t<li>Plot the ordered pairs.<\/li>\r\n \t<li>Connect the points if they form a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\r\nGraph the equation [latex]y=-x+2[\/latex] by plotting points.\r\n[reveal-answer q=\"792137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792137\"]\r\n\r\nFirst, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em>\r\n<table 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).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=-x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n<td>[latex]\\left(-5,7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n<td>[latex]\\left(-3,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(-1,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(1,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n<td>[latex]\\left(3,-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n<td>[latex]\\left(5,-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, plot the points. Connect them if they form a line.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConstruct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].\r\n[reveal-answer q=\"811886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811886\"]\r\n<table 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).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}x+2[\/latex]<\/td>\r\n<td>[latex]\\left(x,y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-2\\right)+2=1[\/latex]<\/td>\r\n<td>[latex]\\left(-2,1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(-1\\right)+2=\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(-1,\\frac{3}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(0\\right)+2=2[\/latex]<\/td>\r\n<td>[latex]\\left(0,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(1\\right)+2=\\frac{5}{2}[\/latex]<\/td>\r\n<td>[latex]\\left(1,\\frac{5}{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y=\\frac{1}{2}\\left(2\\right)+2=3[\/latex]<\/td>\r\n<td>[latex]\\left(2,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/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).\" width=\"487\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]110939[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use Intercepts to Graph Equations<\/h2>\r\nThe <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.\r\n\r\nTo 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].\r\n\r\nTo find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}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{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nTo find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill &amp; \\hfill \\\\ y=3\\left(0\\right)-1\\hfill &amp; \\hfill \\\\ y=-1\\hfill &amp; \\hfill \\\\ \\left(0,-1\\right)\\hfill &amp; y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nWe can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/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\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, find the intercepts<\/h3>\r\n<ol>\r\n \t<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\r\n \t<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\r\nFind the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.\r\n\r\n[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814560\"]\r\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nSet [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nPlot both points and draw a line passing through them.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].\r\n\r\n[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80464\"]\r\n\r\n<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/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.\" width=\"487\" height=\"447\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]92757[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Determine Where Two Functions Intersect<\/span>\r\n\r\n<\/div>\r\nTo determine where two functions intersect, set them equal to each other and solve for\u00a0[latex]x[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining Where Two Functions Intersect<\/h3>\r\nFind the points of intersection of the functions [latex]y=x+2[\/latex] and [latex]y=x^2+3x+2[\/latex].\r\n\r\n[reveal-answer q=\"133750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133750\"]\r\n\r\nSet the two functions equal to each other.\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]x+2=x^2+3x+2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]0=x^2+2x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x^2+2x=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x(x+2)=0[\/latex]<\/p>\r\nSetting [latex]x=0[\/latex] gives us an intersection point at [latex]x=0[\/latex]. To find the corresponding\u00a0<em>y<\/em>-value of the point, let\u00a0[latex]x=0[\/latex] in either function equation: [latex]y=x+2=0+2=2[\/latex].\r\n\r\nSetting [latex]x+2=0[\/latex] gives us an intersection point at [latex]x=-2[\/latex].\u00a0To find the corresponding\u00a0<em>y<\/em>-value of the point, let\u00a0[latex]x=-2[\/latex] in either function equation: [latex]y=x+2=-2+2=0[\/latex].\r\n\r\nNotice that graphically, we can see that the line and the parabola intersect at the points [latex](0,2)[\/latex] and\u00a0[latex](-2,0)[\/latex].\r\n<p style=\"text-align: center;\"><img class=\"alignnone size-medium wp-image-469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5604\/2021\/05\/20220158\/Intersect_1-300x215.jpg\" alt=\"The graph of a parabola and line intersecting at (-2,0) and (0,2)\" width=\"300\" height=\"215\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the points of intersection of the functions [latex]y=x-2[\/latex] and [latex]y=2x^2-4x+1[\/latex].\r\n\r\n[reveal-answer q=\"133751\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133751\"]\r\n<p style=\"text-align: left;\">[latex](1,-1)[\/latex] and\u00a0[latex](\\dfrac{3}{2}, -\\dfrac{1}{2})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph horizontal and vertical lines<\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Graph linear equations in different forms using ordered pairs&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4610,&quot;4&quot;:[null,2,13624051],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Graph linear equations in different forms using ordered pairs<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Graph a linear equation using x and y-intercepts&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4610,&quot;4&quot;:[null,2,13624051],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Graph a linear equation using x and y-intercepts<\/span><\/li>\n<li>Determine if and where two equations intersect<\/li>\n<\/ul>\n<\/div>\n<p>In the first three sections of Module 6, it is very important that we graph the equations we are given so there is a visual to assist us with setting up the needed integral. First, we will review the special equation forms for horizontal and vertical lines. Then, we will review how to graph equations using points and intercepts and determine where two functions intersect.<\/p>\n<h2>Graph Vertical and Horizontal Lines<\/h2>\n<p>The equation of a <strong>vertical line<\/strong> is given as<\/p>\n<div style=\"text-align: center;\">[latex]x=c[\/latex]<\/div>\n<p>where <em>c <\/em>is a constant. The slope of a vertical line is undefined, and regardless of the <em>y-<\/em>value of any point on the line, the <em>x-<\/em>coordinate of the point will be <em>c<\/em>.<\/p>\n<p>Suppose that we want to find the equation of a line containing the following points: [latex]\\left(-3,-5\\right),\\left(-3,1\\right),\\left(-3,3\\right)[\/latex], and [latex]\\left(-3,5\\right)[\/latex]. First, we will find the slope.<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{5 - 3}{-3-\\left(-3\\right)}=\\frac{2}{0}[\/latex]<\/div>\n<p>Zero in the denominator means that the slope is undefined and, therefore, we cannot use point-slope form. However, we can plot the points. Notice that all of the <em>x-<\/em>coordinates are the same and we find a vertical line through [latex]x=-3[\/latex].<\/p>\n<p>The equation of a <strong>horizontal line<\/strong> is given as<\/p>\n<div style=\"text-align: center;\">[latex]y=c[\/latex]<\/div>\n<p>where <em>c <\/em>is a constant. The slope of a horizontal line is zero, and for any <em>x-<\/em>value of a point on the line, the <em>y-<\/em>coordinate will be <em>c<\/em>.<\/p>\n<p>Suppose we want to find the equation of a line that contains the following set of points: [latex]\\left(-2,-2\\right),\\left(0,-2\\right),\\left(3,-2\\right)[\/latex], and [latex]\\left(5,-2\\right)[\/latex]. We can use point-slope form. First, we find the slope using any two points on the line.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m=\\frac{-2-\\left(-2\\right)}{0-\\left(-2\\right)}\\hfill \\\\ =\\frac{0}{2}\\hfill \\\\ =0\\hfill \\end{array}[\/latex]<\/div>\n<p>Use any point for [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] in the formula, or use the <em>y<\/em>-intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y-\\left(-2\\right)=0\\left(x - 3\\right)\\hfill \\\\ y+2=0\\hfill \\\\ y=-2\\hfill \\end{array}[\/latex]<\/div>\n<p>The graph is a horizontal line through [latex]y=-2[\/latex]. Notice that all of the <em>y-<\/em>coordinates are the same.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185925\/CNX_CAT_Figure_02_02_003.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 7 to 4 and the y-axis ranging from negative 4 to 4. The function y = negative 2 and the line x = negative 3 are plotted.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\">The line <i>x<\/i> = \u22123 is a vertical line. The line <i>y<\/i> = \u22122 is a horizontal line.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing Horizontal and Vertical Lines<\/h3>\n<p>Graph the line [latex]x=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q122244\">Show Solution<\/span><\/p>\n<div id=\"q122244\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since [latex]x=[\/latex] is the equation form for a vertical line, we draw a vertical line that goes through 2 on the\u00a0<em>x<\/em>-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-468 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5604\/2021\/05\/20214008\/Vertical-Line_a-300x281.jpg\" alt=\"A vertical line that goes through a value of 2 on the x-axis\" width=\"300\" height=\"281\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm222262\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=222262&theme=oea&iframe_resize_id=ohm222262&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm222263\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=222263&theme=oea&iframe_resize_id=ohm222263&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use Points to Graph Equations<\/h2>\n<div>\n<p>We can plot a set of points to represent an equation. 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\u00a0<\/em>and <em>y-<\/em>values is an ordered pair that can be plotted. The table below\u00a0lists values of <em>x<\/em> from \u20133 to 3 and the resulting values for <em>y<\/em>.<\/p>\n<table 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>We can plot these points from the table. The points for this particular equation form a line, so we can connect them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042408\/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\" \/><\/p>\n<p>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 class=\"textbox\">\n<h3>How To: Given an equation, graph by plotting points<\/h3>\n<ol>\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 graphing easier.<\/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 class=\"textbox exercises\">\n<h3>Example: Graphing an Equation in Two Variables by Plotting Points<\/h3>\n<p>Graph the equation [latex]y=-x+2[\/latex] by plotting points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792137\">Show Solution<\/span><\/p>\n<div id=\"q792137\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we construct a table similar to the one below. Choose <em>x<\/em> values and calculate <em>y.<\/em><\/p>\n<table 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>Now, plot the points. Connect them if they form a line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042411\/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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Construct a table and graph the equation by plotting points: [latex]y=\\frac{1}{2}x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q811886\">Show Solution<\/span><\/p>\n<div id=\"q811886\" class=\"hidden-answer\" style=\"display: none\">\n<table 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><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042413\/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).\" width=\"487\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm110939\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110939&theme=oea&iframe_resize_id=ohm110939&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use Intercepts to Graph Equations<\/h2>\n<p>The <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p>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>To find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}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{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>To find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill & \\hfill \\\\ y=3\\left(0\\right)-1\\hfill & \\hfill \\\\ y=-1\\hfill & \\hfill \\\\ \\left(0,-1\\right)\\hfill & y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/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\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, find the intercepts<\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: 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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814560\">Show Solution<\/span><\/p>\n<div id=\"q814560\" class=\"hidden-answer\" style=\"display: none\">\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Set [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points and draw a line passing through them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80464\">Show Solution<\/span><\/p>\n<div id=\"q80464\" class=\"hidden-answer\" style=\"display: none\">\n<p><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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/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.\" width=\"487\" height=\"447\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92757\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92757&theme=oea&iframe_resize_id=ohm92757&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Determine Where Two Functions Intersect<\/span><\/p>\n<\/div>\n<p>To determine where two functions intersect, set them equal to each other and solve for\u00a0[latex]x[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Determining Where Two Functions Intersect<\/h3>\n<p>Find the points of intersection of the functions [latex]y=x+2[\/latex] and [latex]y=x^2+3x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133750\">Show Solution<\/span><\/p>\n<div id=\"q133750\" class=\"hidden-answer\" style=\"display: none\">\n<p>Set the two functions equal to each other.<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]x+2=x^2+3x+2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0=x^2+2x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x^2+2x=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x(x+2)=0[\/latex]<\/p>\n<p>Setting [latex]x=0[\/latex] gives us an intersection point at [latex]x=0[\/latex]. To find the corresponding\u00a0<em>y<\/em>-value of the point, let\u00a0[latex]x=0[\/latex] in either function equation: [latex]y=x+2=0+2=2[\/latex].<\/p>\n<p>Setting [latex]x+2=0[\/latex] gives us an intersection point at [latex]x=-2[\/latex].\u00a0To find the corresponding\u00a0<em>y<\/em>-value of the point, let\u00a0[latex]x=-2[\/latex] in either function equation: [latex]y=x+2=-2+2=0[\/latex].<\/p>\n<p>Notice that graphically, we can see that the line and the parabola intersect at the points [latex](0,2)[\/latex] and\u00a0[latex](-2,0)[\/latex].<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5604\/2021\/05\/20220158\/Intersect_1-300x215.jpg\" alt=\"The graph of a parabola and line intersecting at (-2,0) and (0,2)\" width=\"300\" height=\"215\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the points of intersection of the functions [latex]y=x-2[\/latex] and [latex]y=2x^2-4x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133751\">Show Solution<\/span><\/p>\n<div id=\"q133751\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex](1,-1)[\/latex] and\u00a0[latex](\\dfrac{3}{2}, -\\dfrac{1}{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1976\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen 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