{"id":3765,"date":"2021-05-12T20:24:06","date_gmt":"2021-05-12T20:24:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/plot-points\/"},"modified":"2021-05-12T20:24:06","modified_gmt":"2021-05-12T20:24:06","slug":"plot-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/plot-points\/","title":{"raw":"Skills Review for Review of Functions","rendered":"Skills Review for Review of Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Plot points on a rectangular coordinate system<\/li>\n \t<li>Identify the x- and y-intercepts from the equation of a line<\/li>\n \t<li>Multiply binomials - FOIL<\/li>\n<\/ul>\n<\/div>\nSome of the topics covered in the Review for Functions section include drawing the graph of a function, using intercepts to graph a function, and performing operations on functions. How to plot points in the Cartesian coordinate system is reviewed here to prepare you for graphing functions. To prepare you for using intercepts to graph a function, how to find intercepts of functions is also reviewed. Finally, how to square a binomial will be discussed; this will come in handy when performing operations on functions.\n<h2>Plot Points<\/h2>\nThe 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 <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.\n\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\/12042358\/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\"> <b>The Cartesian coordinate system with all four quadrants labeled.<\/b>[\/caption]\n\nThe center of the plane is the point at which the two axes cross. It is known as the <strong>origin&nbsp;<\/strong>or point [latex]\\left(0,0\\right)[\/latex].\n\nEach point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,&nbsp;or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> 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&nbsp;<\/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.\n\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\/12042403\/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\"> <b>An illustration of how to plot the point (3,-1).<\/b>[\/caption]\n\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Cartesian Coordinate System<\/h3>\n\n<hr>\n\nA two-dimensional plane where the\n<ul>\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>\nA 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.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\nPlot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.\n[reveal-answer q=\"380739\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"380739\"]\n\nTo 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.\n\nTo 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.\n\nTo 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.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042406\/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.\" width=\"487\" height=\"442\">\n<h4>Analysis of the Solution<\/h4>\nNote 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.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]92753[\/ohm_question]\n\n<\/div>\nhttps:\/\/www.youtube.com\/watch?v=7JMXi_FxA2o\n<h2>Find Intercepts<\/h2>\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.\n\nRegardless of the type of equation or function we are dealing with, 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].\n\nTo find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].\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>\nTo find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].\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>\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.\n\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\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Given an equation, find the intercepts<\/h3>\n\n<hr>\n\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: 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[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"814560\"]\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.\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>\nSet [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.\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>\nPlot both points and draw a line passing through them.\n\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\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nFind the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].\n\n[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"80464\"]\n\n<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\n\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\">\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Square Binomials<\/h2>\nWhen a binomial is squared, the result is called a <strong>perfect square trinomial<\/strong>. We can square a binomial by multiplying the binomial by itself. However, there is always a certain form that results from squaring a binomial. Memorizing the form makes squaring binomials much easier. Let\u2019s look at a few binomials squared to familiarize ourselves with the form.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x - 3\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x - 1\\right)}^{2}&amp; =&amp; 4{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\nNotice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Squaring a Binomial<\/h3>\n\n<hr>\n\nWhen a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.\n<div style=\"text-align: center;\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<div>\n\n&nbsp;\n\nGiven a binomial, square it using the <strong>Binomial Squaring Shortcut:<\/strong>\n<ol>\n \t<li>Square the first term of the binomial.<\/li>\n \t<li>Square the last term of the binomial.<\/li>\n \t<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n \t<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Expanding Perfect Squares<\/h3>\nExpand [latex]{\\left(3x - 8\\right)}^{2}[\/latex].\n\n[reveal-answer q=\"733978\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"733978\"]\n\nBegin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.\n<div style=\"text-align: center;\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/div>\n<p style=\"text-align: center;\">[latex]9{x}^{2}-48x+64[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nExpand\n\na. [latex]{\\left(4x - 1\\right)}^{2}[\/latex]\n\nb.&nbsp;[latex]{\\left(5y + 3\\right)}^{2}[\/latex]\n\nc.&nbsp;[latex]{\\left(w + 4z\\right)}^{2}[\/latex]\n\n[reveal-answer q=\"278544\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"278544\"]\n\na. [latex]16{x}^{2}-8x+1[\/latex]\n\nb. [latex]25{y}^{2}+30y+9[\/latex]\n\nc.&nbsp;[latex]w^{2}+8wz+16z^{2}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]1825[\/ohm_question]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Plot points on a rectangular coordinate system<\/li>\n<li>Identify the x- and y-intercepts from the equation of a line<\/li>\n<li>Multiply binomials &#8211; FOIL<\/li>\n<\/ul>\n<\/div>\n<p>Some of the topics covered in the Review for Functions section include drawing the graph of a function, using intercepts to graph a function, and performing operations on functions. How to plot points in the Cartesian coordinate system is reviewed here to prepare you for graphing functions. To prepare you for using intercepts to graph a function, how to find intercepts of functions is also reviewed. Finally, how to square a binomial will be discussed; this will come in handy when performing operations on functions.<\/p>\n<h2>Plot Points<\/h2>\n<p>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 <strong>quadrant<\/strong>; the quadrants are numbered counterclockwise as shown in the figure below.<\/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\/12042358\/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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>The Cartesian coordinate system with all four quadrants labeled.<\/b><\/p>\n<\/div>\n<p>The center of the plane is the point at which the two axes cross. It is known as the <strong>origin&nbsp;<\/strong>or point [latex]\\left(0,0\\right)[\/latex].<\/p>\n<p>Each point in the plane is identified by its <strong><em>x-<\/em>coordinate<\/strong>,&nbsp;or horizontal displacement from the origin, and its <strong><em>y-<\/em>coordinate<\/strong>, or vertical displacement from the origin. Together we write them as an <strong>ordered pair<\/strong> 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&nbsp;<\/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.<\/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\/12042403\/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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>An illustration of how to plot the point (3,-1).<\/b><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Cartesian Coordinate System<\/h3>\n<hr \/>\n<p>A two-dimensional plane where the<\/p>\n<ul>\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>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 class=\"textbox exercises\">\n<h3>Example: Plotting Points in a Rectangular Coordinate System<\/h3>\n<p>Plot the points [latex]\\left(-2,4\\right)[\/latex], [latex]\\left(3,3\\right)[\/latex], and [latex]\\left(0,-3\\right)[\/latex] in the coordinate plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380739\">Show Solution<\/span><\/p>\n<div id=\"q380739\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>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>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.<\/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\/12042406\/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.\" width=\"487\" height=\"442\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92753\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92753&theme=oea&iframe_resize_id=ohm92753&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Plotting Points on the Coordinate Plane\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7JMXi_FxA2o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Find Intercepts<\/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>Regardless of the type of equation or function we are dealing with, 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 shaded\">\n<h3 style=\"text-align: center;\">Given an equation, find the intercepts<\/h3>\n<hr \/>\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<h2>Square Binomials<\/h2>\n<p>When a binomial is squared, the result is called a <strong>perfect square trinomial<\/strong>. We can square a binomial by multiplying the binomial by itself. However, there is always a certain form that results from squaring a binomial. Memorizing the form makes squaring binomials much easier. Let\u2019s look at a few binomials squared to familiarize ourselves with the form.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}& =& \\text{ }{x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x - 3\\right)}^{2}& =& \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x - 1\\right)}^{2}& =& 4{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\n<p>Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Squaring a Binomial<\/h3>\n<hr \/>\n<p>When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>Given a binomial, square it using the <strong>Binomial Squaring Shortcut:<\/strong><\/p>\n<ol>\n<li>Square the first term of the binomial.<\/li>\n<li>Square the last term of the binomial.<\/li>\n<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Expanding Perfect Squares<\/h3>\n<p>Expand [latex]{\\left(3x - 8\\right)}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q733978\">Show Solution<\/span><\/p>\n<div id=\"q733978\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/div>\n<p style=\"text-align: center;\">[latex]9{x}^{2}-48x+64[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Expand<\/p>\n<p>a. [latex]{\\left(4x - 1\\right)}^{2}[\/latex]<\/p>\n<p>b.&nbsp;[latex]{\\left(5y + 3\\right)}^{2}[\/latex]<\/p>\n<p>c.&nbsp;[latex]{\\left(w + 4z\\right)}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278544\">Show Solution<\/span><\/p>\n<div id=\"q278544\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. [latex]16{x}^{2}-8x+1[\/latex]<\/p>\n<p>b. [latex]25{y}^{2}+30y+9[\/latex]<\/p>\n<p>c.&nbsp;[latex]w^{2}+8wz+16z^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1825\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1825&theme=oea&iframe_resize_id=ohm1825&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-3765\">\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":17533,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"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 Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3765","chapter","type-chapter","status-publish","hentry"],"part":3764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3765","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3765\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/3764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3765\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3765"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3765"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3765"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}