{"id":750,"date":"2025-07-15T15:12:21","date_gmt":"2025-07-15T15:12:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=750"},"modified":"2026-03-18T03:17:53","modified_gmt":"2026-03-18T03:17:53","slug":"graphs-of-linear-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-linear-functions-fresh-take\/","title":{"raw":"Graphs of Linear Functions: Fresh Take","rendered":"Graphs of Linear Functions: Fresh Take"},"content":{"raw":"<section id=\"fs-id1165137543411\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph linear functions using a table of values.<\/li>\r\n \t<li>Identify and interpret the slope and intercepts of a linear function.<\/li>\r\n \t<li>Write the equation for a linear function from the graph of a line.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Graphing a Linear Function Using Transformations<\/h2>\r\n<p id=\"fs-id1165137695235\">Another option for graphing is to use <strong>transformations<\/strong> of the identity function [latex]f(x)=x[\/latex] . A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.<\/p>\r\n\r\n<section id=\"fs-id1165137662254\">\r\n<h3>Vertical Stretch or Compression<\/h3>\r\n<p id=\"fs-id1165137444518\">In the equation [latex]f(x)=mx[\/latex], the <em>m<\/em>\u00a0is acting as the <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the identity function. When <em>m<\/em>\u00a0is negative, there is also a vertical reflection of the graph. Notice in Figure 4\u00a0that multiplying the equation of [latex]f(x)=x[\/latex] by <em>m<\/em>\u00a0stretches the graph of <i>f<\/i>\u00a0by a factor of <em>m<\/em>\u00a0units if <em>m\u00a0<\/em>&gt; 1 and compresses the graph of <em>f<\/em>\u00a0by a factor of <em>m<\/em>\u00a0units if 0 &lt; <em>m\u00a0<\/em>&lt; 1. This means the larger the absolute value of <em>m<\/em>, the steeper the slope.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010645\/CNX_Precalc_Figure_02_02_0052.jpg\" alt=\"Graph with several linear functions including y = 3x, y = 2x, y = x, y = (1\/2)x, y = (1\/3)x, y = (-1\/2)x, y = -x, and y = -2x\" width=\"900\" height=\"759\" \/>\r\n\r\n<\/section><\/section>\r\n<p style=\"text-align: center;\"><strong>Figure 4.<\/strong> Vertical stretches and compressions and reflections on the function [latex]f\\left(x\\right)=x[\/latex].<\/p>\r\n\r\n<section id=\"fs-id1165137543411\"><section id=\"fs-id1165135667863\">\r\n<h3>Vertical Shift<\/h3>\r\n<p id=\"fs-id1165137600044\">In [latex]f\\left(x\\right)=mx+b[\/latex], the <em>b<\/em>\u00a0acts as the <strong>vertical shift<\/strong>, moving the graph up and down without affecting the slope of the line. Notice in Figure 5\u00a0that adding a value of <em>b<\/em>\u00a0to the equation of [latex]f\\left(x\\right)=x[\/latex] shifts the graph of\u00a0<em>f<\/em>\u00a0a total of <em>b<\/em>\u00a0units up if <em>b<\/em>\u00a0is positive and\u00a0|<em>b<\/em>| units down if <em>b<\/em>\u00a0is negative.<\/p>\r\n<span id=\"fs-id1165137634286\"> <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010645\/CNX_Precalc_Figure_02_02_0062.jpg\" alt=\"graph showing y = x , y = x+2, y = x+4, y = x-2, y = x-4\" width=\"900\" height=\"759\" \/><\/span>\r\n\r\n<\/section><\/section>\r\n<p style=\"text-align: center;\"><strong>Figure 5.<\/strong> This graph illustrates vertical shifts of the function [latex]f(x)=x[\/latex].<\/p>\r\n\r\n<section id=\"fs-id1165137543411\"><section id=\"fs-id1165135667863\">\r\n<p id=\"fs-id1165137564772\">Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.<\/p>\r\n\r\n<div id=\"fs-id1165137641217\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137680349\">How To: Given the equation of a linear function, use transformations to graph the linear function in the form [latex]f\\left(x\\right)=mx+b[\/latex].<\/h3>\r\n<ol id=\"fs-id1165135449594\">\r\n \t<li>Graph [latex]f\\left(x\\right)=x[\/latex].<\/li>\r\n \t<li>Vertically stretch or compress the graph by a factor <em>m<\/em>.<\/li>\r\n \t<li>Shift the graph up or down <em>b<\/em>\u00a0units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_02_02_03\" class=\"example\">\r\n<div id=\"fs-id1165137456438\" class=\"exercise\"><section class=\"textbox example\" aria-label=\"Example\">\r\n<h3>Example 3: Graphing by Using Transformations<\/h3>\r\n<p id=\"fs-id1165135570273\">Graph [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex] using transformations.<\/p>\r\n[reveal-answer q=\"653846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"653846\"]\r\n<p id=\"fs-id1165135192082\">The equation for the function shows that [latex]m=\\frac{1}{2}[\/latex] so the identity function is vertically compressed by [latex]\\frac{1}{2}[\/latex]. The equation for the function also shows that <em>b\u00a0<\/em>= \u20133 so the identity function is vertically shifted down 3 units. First, graph the identity function, and show the vertical compression.<\/p>\r\n<span id=\"fs-id1165135245753\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010645\/CNX_Precalc_Figure_02_02_0072.jpg\" alt=\"graph showing the lines y = x and y = (1\/2)x\" width=\"487\" height=\"378\" \/><\/span>\r\n<p style=\"text-align: center;\"><strong>Figure 6.\u00a0<\/strong>The function, <em>y\u00a0<\/em>= <em>x<\/em>, compressed by a factor of [latex]\\frac{1}{2}[\/latex].<\/p>\r\nThen show the vertical shift.\r\n<p style=\"text-align: center;\"><span id=\"fs-id1165137610735\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010646\/CNX_Precalc_Figure_02_02_0082.jpg\" alt=\"Graph showing the lines y = (1\/2)x, and y = (1\/2) + 3\" width=\"487\" height=\"377\" \/><\/span>\r\n<strong>Figure 7.<\/strong> The function [latex]y=\\frac{1}{2}x[\/latex], shifted down 3 units.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p id=\"fs-id1165137823624\">Graph [latex]f\\left(x\\right)=4+2x[\/latex], using transformations.<\/p>\r\n[reveal-answer q=\"713974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"713974\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010646\/CNX_Precalc_Figure_02_02_0092.jpg\" alt=\"\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135176280\" class=\"note precalculus qa textbox\">\r\n<h3 id=\"fs-id1165137603576\">Q &amp; A<\/h3>\r\n<strong>In Example 3, could we have sketched the graph by reversing the order of the transformations?<\/strong>\r\n<p id=\"fs-id1165137730398\"><em>No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first. For example, following the order: Let the input be 2.<\/em><\/p>\r\n\r\n<div id=\"fs-id1165137619677\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}f(2)&amp;=\\frac{1}{2}(2)-3 \\\\ &amp;=1-3 \\\\ &amp;=-2 \\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<h2>Writing the Equation for a Function from the Graph of a Line<\/h2>\r\n<section id=\"fs-id1165137531122\">\r\n<p id=\"fs-id1165135408570\">Recall that in Linear Functions, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at Figure 8. We can see right away that the graph crosses the <em>y<\/em>-axis at the point (0, 4) so this is the <em>y<\/em>-intercept.<span id=\"fs-id1165137629251\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"369\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010646\/CNX_Precalc_Figure_02_02_0102.jpg\" alt=\"\" width=\"369\" height=\"378\" \/> <b>Figure 8<\/b>[\/caption]\r\n<p id=\"fs-id1165135501156\">Then we can calculate the slope by finding the rise and run. We can choose any two points, but let\u2019s look at the point (\u20132, 0). To get from this point to the <em>y-<\/em>intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be<\/p>\r\n\r\n<div id=\"fs-id1165137526424\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex]<\/div>\r\n<p id=\"fs-id1165135684358\">Substituting the slope and <em>y-<\/em>intercept into the slope-intercept form of a line gives<\/p>\r\n\r\n<div id=\"fs-id1165135316180\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=2x+4[\/latex]<\/div>\r\n<div id=\"fs-id1165137836529\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137760034\">How To: Given a graph of linear function, find the equation to describe the function.<\/h3>\r\n<ol id=\"fs-id1165137769882\">\r\n \t<li>Identify the <em>y-<\/em>intercept of an equation.<\/li>\r\n \t<li>Choose two points to determine the slope.<\/li>\r\n \t<li>Substitute the <em>y-<\/em>intercept and slope into the slope-intercept form of a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<h3>Example 4: Matching Linear Functions to Their Graphs<\/h3>\r\n<b>Figure 10<\/b>[\/hidden-answer]\r\n\r\n<\/section><\/section>\r\n<figure class=\"small\"><\/figure>\r\n<section id=\"fs-id1165137767695\">\r\n<h2>Finding the <em>x<\/em>-intercept of a Line<\/h2>\r\n<p id=\"fs-id1165137665075\">So far, we have been finding the <em>y-<\/em>intercepts of a function: the point at which the graph of the function crosses the <em>y<\/em>-axis. A function may also have an <strong><em>x<\/em><\/strong><strong>-intercept,<\/strong> which is the <em>x<\/em>-coordinate of the point where the graph of the function crosses the <em>x<\/em>-axis. In other words, it is the input value when the output value is zero.<\/p>\r\n<p id=\"fs-id1165135528375\">To find the <em>x<\/em>-intercept, set a function <em>f<\/em>(<em>x<\/em>) equal to zero and solve for the value of <em>x<\/em>. For example, consider the function shown.<\/p>\r\n\r\n<div id=\"eip-901\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=3x - 6[\/latex]<\/div>\r\n<p id=\"fs-id1165137549960\">Set the function equal to 0 and solve for <em>x<\/em>.<\/p>\r\n\r\n<div id=\"fs-id1165137595415\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&amp;0=3x - 6 \\\\ &amp;6=3x \\\\ &amp;2=x \\\\ &amp;x=2 \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165135149818\">The graph of the function crosses the <em>x<\/em>-axis at the point (2, 0).<\/p>\r\n\r\n<div id=\"fs-id1165137705101\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165137705106\"><strong>Do all linear functions have <em>x<\/em>-intercepts?<\/strong><\/p>\r\n<p id=\"fs-id1165137827599\"><em>No. However, linear functions of the form <\/em>y\u00a0<em>= <\/em>c<em>, where <\/em>c<em> is a nonzero real number are the only examples of linear functions with no <\/em>x<em>-intercept. For example, <\/em>y\u00a0<em>= 5 is a horizontal line 5 units above the <\/em>x<em>-axis. This function has no <\/em>x<em>-intercepts<\/em>.<\/p>\r\n\r\n<figure id=\"CNX_Precalc_Figure_02_02_026\" class=\"medium\"><\/figure>\r\n<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"421\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010647\/CNX_Precalc_Figure_02_02_0262.jpg\" alt=\"Graph of y = 5.\" width=\"421\" height=\"231\" \/> <b>Figure 11<\/b>[\/caption]\r\n\r\n<div id=\"fs-id1165137653298\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: <em>x<\/em>-intercept<\/h3>\r\n<p id=\"fs-id1165137663549\">The <strong><em>x<\/em>-intercept<\/strong> of the function is the point where the graph crosses the\u00a0<em>x<\/em>-axis. Points on the <em>x<\/em>-axis have the form (<em>x<\/em>,0) so we can find <em>x<\/em>-intercepts by setting\u00a0<em>f<\/em>(<em>x<\/em>) = 0. For a linear function, we solve the equation <em>mx\u00a0<\/em>+ <em>b\u00a0<\/em>= 0<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<h3>Example 5: Finding an <em>x<\/em>-intercept<\/h3>\r\n<p id=\"fs-id1165137663560\">Find the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].<\/p>\r\n[reveal-answer q=\"383732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383732\"]\r\n<p id=\"fs-id1165137424379\">Set the function equal to zero to solve for <em>x<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}0&amp;=\\frac{1}{2}x - 3\\\\[1mm] 3&amp;=\\frac{1}{2}x\\\\[1mm] 6&amp;=x\\\\[1mm] x&amp;=6\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137415633\">The graph crosses the <em>x<\/em>-axis at the point (6,0).<\/p>\r\n\r\n<div id=\"Example_02_02_05\" class=\"example\">\r\n<div id=\"fs-id1165137805711\" class=\"exercise\">\r\n<div id=\"fs-id1165135450383\" class=\"commentary\">\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135450388\">A graph of the function is shown in Figure 12. We can see that the <em>x<\/em>-intercept is (6, 0) as we expected.<\/p>\r\n<span id=\"fs-id1165137424274\"> <img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010648\/CNX_Precalc_Figure_02_02_0132.jpg\" alt=\"\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137727385\" class=\"note precalculus try\">\r\n<div id=\"ti_02_02_04\" class=\"exercise\">\r\n<div id=\"fs-id1165134389962\" class=\"problem\">\r\n<p style=\"text-align: center;\"><strong>Figure 12.\u00a0<\/strong>The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p id=\"fs-id1165134389964\">Find the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{4}x - 4[\/latex].<\/p>\r\n[reveal-answer q=\"946091\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"946091\"]\r\n\r\n[latex]\\left(16,\\text{ 0}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section><\/section><\/section>\r\n<h2 id=\"fs-id1165137410528\">Calculating and Interpreting Slope<\/h2>\r\n<p id=\"fs-id1165137573268\">In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the <strong>slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex]\u00a0and [latex]{y}_{2}[\/latex] \u2014which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex]\u2014we can calculate the slope [latex]m[\/latex],\u00a0as follows<\/p>\r\n\r\n<div id=\"fs-id1165137757690\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[latex][\/latex]<\/div>\r\n<p id=\"fs-id1165137767606\">where [latex]\\Delta y[\/latex] is the vertical displacement and [latex]\\Delta x[\/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] for the function [latex]f[\/latex], [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex], so we could equivalently write<\/p>\r\n\r\n<div id=\"fs-id1165137438737\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[latex][\/latex]<\/div>\r\nThe graph in Figure 5\u00a0indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex]\r\nand [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex],\u00a0is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010639\/CNX_Precalc_Figure_02_01_005n2.jpg\" alt=\"Graph depicting how to calculate the slope of a line\" width=\"487\" height=\"569\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n<em>The slope of a function is calculated by the change in [latex]y[\/latex] <\/em>\r\n<div id=\"Example_02_01_03\" class=\"example\">\r\n<div id=\"fs-id1165137641892\" class=\"exercise\"><section class=\"textbox example\" aria-label=\"Example\">\r\n<h3>Finding the Slope of a Linear Function<\/h3>\r\n<p id=\"fs-id1165137724908\">If [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function, and [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\r\n[reveal-answer q=\"603714\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"603714\"]\r\n<p id=\"fs-id1165137714054\">The coordinate pairs are [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]. To find the rate of change, we divide the change in output by the change in input.<\/p>\r\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-\\left(-2\\right)}{8 - 3}=\\frac{3}{5}[\/latex]<\/p>\r\n<p id=\"fs-id1165137469338\">We could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m&gt;0[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137842413\">As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or <em>y<\/em>-coordinate, used corresponds with the first input value, or <em>x<\/em>-coordinate, used.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p id=\"fs-id1165137451238\">If [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function, and [latex]\\left(2,\\text{ }3\\right)[\/latex]\u00a0and [latex]\\left(0,\\text{ }4\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\r\n[reveal-answer q=\"786191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"786191\"]\r\n\r\n[latex]m=\\frac{4 - 3}{0 - 2}=\\frac{1}{-2}=-\\frac{1}{2}[\/latex] ; decreasing because [latex]m&lt;0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]169760[\/ohm_question]<\/section><\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox example\" aria-label=\"Example\">\r\n<h3>Example 4: Finding the Population Change from a Linear Function<\/h3>\r\n<p id=\"fs-id1165135386495\">The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.<\/p>\r\n[reveal-answer q=\"870587\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"870587\"]\r\n<p id=\"fs-id1165137415969\">The rate of change relates the change in population to the change in time. The population increased by [latex]27,800 - 23,400=4400[\/latex] people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{4,400\\text{ people}}{4\\text{ years}}=1,100\\text{ }\\frac{\\text{people}}{\\text{year}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex][\/latex]<\/p>\r\n<p id=\"fs-id1165137705484\">So the population increased by 1,100 people per year.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137451439\">Because we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.[reveal-answer q=\"468046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"468046\"][latex]m=\\frac{1,868 - 1,442}{2,012 - 2,009}=\\frac{426}{3}=142\\text{ people per year}[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question]57147[\/ohm_question]<\/section><\/div>","rendered":"<section id=\"fs-id1165137543411\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph linear functions using a table of values.<\/li>\n<li>Identify and interpret the slope and intercepts of a linear function.<\/li>\n<li>Write the equation for a linear function from the graph of a line.<\/li>\n<\/ul>\n<\/section>\n<h2>Graphing a Linear Function Using Transformations<\/h2>\n<p id=\"fs-id1165137695235\">Another option for graphing is to use <strong>transformations<\/strong> of the identity function [latex]f(x)=x[\/latex] . A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.<\/p>\n<section id=\"fs-id1165137662254\">\n<h3>Vertical Stretch or Compression<\/h3>\n<p id=\"fs-id1165137444518\">In the equation [latex]f(x)=mx[\/latex], the <em>m<\/em>\u00a0is acting as the <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the identity function. When <em>m<\/em>\u00a0is negative, there is also a vertical reflection of the graph. Notice in Figure 4\u00a0that multiplying the equation of [latex]f(x)=x[\/latex] by <em>m<\/em>\u00a0stretches the graph of <i>f<\/i>\u00a0by a factor of <em>m<\/em>\u00a0units if <em>m\u00a0<\/em>&gt; 1 and compresses the graph of <em>f<\/em>\u00a0by a factor of <em>m<\/em>\u00a0units if 0 &lt; <em>m\u00a0<\/em>&lt; 1. This means the larger the absolute value of <em>m<\/em>, the steeper the slope.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010645\/CNX_Precalc_Figure_02_02_0052.jpg\" alt=\"Graph with several linear functions including y = 3x, y = 2x, y = x, y = (1\/2)x, y = (1\/3)x, y = (-1\/2)x, y = -x, and y = -2x\" width=\"900\" height=\"759\" \/><\/p>\n<\/section>\n<\/section>\n<p style=\"text-align: center;\"><strong>Figure 4.<\/strong> Vertical stretches and compressions and reflections on the function [latex]f\\left(x\\right)=x[\/latex].<\/p>\n<section>\n<section id=\"fs-id1165135667863\">\n<h3>Vertical Shift<\/h3>\n<p id=\"fs-id1165137600044\">In [latex]f\\left(x\\right)=mx+b[\/latex], the <em>b<\/em>\u00a0acts as the <strong>vertical shift<\/strong>, moving the graph up and down without affecting the slope of the line. Notice in Figure 5\u00a0that adding a value of <em>b<\/em>\u00a0to the equation of [latex]f\\left(x\\right)=x[\/latex] shifts the graph of\u00a0<em>f<\/em>\u00a0a total of <em>b<\/em>\u00a0units up if <em>b<\/em>\u00a0is positive and\u00a0|<em>b<\/em>| units down if <em>b<\/em>\u00a0is negative.<\/p>\n<p><span id=\"fs-id1165137634286\"> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010645\/CNX_Precalc_Figure_02_02_0062.jpg\" alt=\"graph showing y = x , y = x+2, y = x+4, y = x-2, y = x-4\" width=\"900\" height=\"759\" \/><\/span><\/p>\n<\/section>\n<\/section>\n<p style=\"text-align: center;\"><strong>Figure 5.<\/strong> This graph illustrates vertical shifts of the function [latex]f(x)=x[\/latex].<\/p>\n<section>\n<section>\n<p id=\"fs-id1165137564772\">Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.<\/p>\n<div id=\"fs-id1165137641217\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137680349\">How To: Given the equation of a linear function, use transformations to graph the linear function in the form [latex]f\\left(x\\right)=mx+b[\/latex].<\/h3>\n<ol id=\"fs-id1165135449594\">\n<li>Graph [latex]f\\left(x\\right)=x[\/latex].<\/li>\n<li>Vertically stretch or compress the graph by a factor <em>m<\/em>.<\/li>\n<li>Shift the graph up or down <em>b<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_02_03\" class=\"example\">\n<div id=\"fs-id1165137456438\" class=\"exercise\">\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3>Example 3: Graphing by Using Transformations<\/h3>\n<p id=\"fs-id1165135570273\">Graph [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex] using transformations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q653846\">Show Solution<\/button><\/p>\n<div id=\"q653846\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135192082\">The equation for the function shows that [latex]m=\\frac{1}{2}[\/latex] so the identity function is vertically compressed by [latex]\\frac{1}{2}[\/latex]. The equation for the function also shows that <em>b\u00a0<\/em>= \u20133 so the identity function is vertically shifted down 3 units. First, graph the identity function, and show the vertical compression.<\/p>\n<p><span id=\"fs-id1165135245753\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010645\/CNX_Precalc_Figure_02_02_0072.jpg\" alt=\"graph showing the lines y = x and y = (1\/2)x\" width=\"487\" height=\"378\" \/><\/span><\/p>\n<p style=\"text-align: center;\"><strong>Figure 6.\u00a0<\/strong>The function, <em>y\u00a0<\/em>= <em>x<\/em>, compressed by a factor of [latex]\\frac{1}{2}[\/latex].<\/p>\n<p>Then show the vertical shift.<\/p>\n<p style=\"text-align: center;\"><span id=\"fs-id1165137610735\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010646\/CNX_Precalc_Figure_02_02_0082.jpg\" alt=\"Graph showing the lines y = (1\/2)x, and y = (1\/2) + 3\" width=\"487\" height=\"377\" \/><\/span><br \/>\n<strong>Figure 7.<\/strong> The function [latex]y=\\frac{1}{2}x[\/latex], shifted down 3 units.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p id=\"fs-id1165137823624\">Graph [latex]f\\left(x\\right)=4+2x[\/latex], using transformations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q713974\">Show Solution<\/button><\/p>\n<div id=\"q713974\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010646\/CNX_Precalc_Figure_02_02_0092.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176280\" class=\"note precalculus qa textbox\">\n<h3 id=\"fs-id1165137603576\">Q &amp; A<\/h3>\n<p><strong>In Example 3, could we have sketched the graph by reversing the order of the transformations?<\/strong><\/p>\n<p id=\"fs-id1165137730398\"><em>No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first. For example, following the order: Let the input be 2.<\/em><\/p>\n<div id=\"fs-id1165137619677\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}f(2)&=\\frac{1}{2}(2)-3 \\\\ &=1-3 \\\\ &=-2 \\end{align}[\/latex]<\/div>\n<\/div>\n<h2>Writing the Equation for a Function from the Graph of a Line<\/h2>\n<section id=\"fs-id1165137531122\">\n<p id=\"fs-id1165135408570\">Recall that in Linear Functions, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at Figure 8. We can see right away that the graph crosses the <em>y<\/em>-axis at the point (0, 4) so this is the <em>y<\/em>-intercept.<span id=\"fs-id1165137629251\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 369px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010646\/CNX_Precalc_Figure_02_02_0102.jpg\" alt=\"\" width=\"369\" height=\"378\" \/><figcaption class=\"wp-caption-text\"><b>Figure 8<\/b><\/figcaption><\/figure>\n<p id=\"fs-id1165135501156\">Then we can calculate the slope by finding the rise and run. We can choose any two points, but let\u2019s look at the point (\u20132, 0). To get from this point to the <em>y-<\/em>intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be<\/p>\n<div id=\"fs-id1165137526424\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex]<\/div>\n<p id=\"fs-id1165135684358\">Substituting the slope and <em>y-<\/em>intercept into the slope-intercept form of a line gives<\/p>\n<div id=\"fs-id1165135316180\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=2x+4[\/latex]<\/div>\n<div id=\"fs-id1165137836529\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137760034\">How To: Given a graph of linear function, find the equation to describe the function.<\/h3>\n<ol id=\"fs-id1165137769882\">\n<li>Identify the <em>y-<\/em>intercept of an equation.<\/li>\n<li>Choose two points to determine the slope.<\/li>\n<li>Substitute the <em>y-<\/em>intercept and slope into the slope-intercept form of a line.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3>Example 4: Matching Linear Functions to Their Graphs<\/h3>\n<p><b>Figure 10<\/b>[\/hidden-answer]<\/p>\n<\/section>\n<\/section>\n<figure class=\"small\"><\/figure>\n<section id=\"fs-id1165137767695\">\n<h2>Finding the <em>x<\/em>-intercept of a Line<\/h2>\n<p id=\"fs-id1165137665075\">So far, we have been finding the <em>y-<\/em>intercepts of a function: the point at which the graph of the function crosses the <em>y<\/em>-axis. A function may also have an <strong><em>x<\/em><\/strong><strong>-intercept,<\/strong> which is the <em>x<\/em>-coordinate of the point where the graph of the function crosses the <em>x<\/em>-axis. In other words, it is the input value when the output value is zero.<\/p>\n<p id=\"fs-id1165135528375\">To find the <em>x<\/em>-intercept, set a function <em>f<\/em>(<em>x<\/em>) equal to zero and solve for the value of <em>x<\/em>. For example, consider the function shown.<\/p>\n<div id=\"eip-901\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=3x - 6[\/latex]<\/div>\n<p id=\"fs-id1165137549960\">Set the function equal to 0 and solve for <em>x<\/em>.<\/p>\n<div id=\"fs-id1165137595415\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&0=3x - 6 \\\\ &6=3x \\\\ &2=x \\\\ &x=2 \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165135149818\">The graph of the function crosses the <em>x<\/em>-axis at the point (2, 0).<\/p>\n<div id=\"fs-id1165137705101\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165137705106\"><strong>Do all linear functions have <em>x<\/em>-intercepts?<\/strong><\/p>\n<p id=\"fs-id1165137827599\"><em>No. However, linear functions of the form <\/em>y\u00a0<em>= <\/em>c<em>, where <\/em>c<em> is a nonzero real number are the only examples of linear functions with no <\/em>x<em>-intercept. For example, <\/em>y\u00a0<em>= 5 is a horizontal line 5 units above the <\/em>x<em>-axis. This function has no <\/em>x<em>-intercepts<\/em>.<\/p>\n<figure id=\"CNX_Precalc_Figure_02_02_026\" class=\"medium\"><\/figure>\n<\/div>\n<figure style=\"width: 421px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010647\/CNX_Precalc_Figure_02_02_0262.jpg\" alt=\"Graph of y = 5.\" width=\"421\" height=\"231\" \/><figcaption class=\"wp-caption-text\"><b>Figure 11<\/b><\/figcaption><\/figure>\n<div id=\"fs-id1165137653298\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: <em>x<\/em>-intercept<\/h3>\n<p id=\"fs-id1165137663549\">The <strong><em>x<\/em>-intercept<\/strong> of the function is the point where the graph crosses the\u00a0<em>x<\/em>-axis. Points on the <em>x<\/em>-axis have the form (<em>x<\/em>,0) so we can find <em>x<\/em>-intercepts by setting\u00a0<em>f<\/em>(<em>x<\/em>) = 0. For a linear function, we solve the equation <em>mx\u00a0<\/em>+ <em>b\u00a0<\/em>= 0<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3>Example 5: Finding an <em>x<\/em>-intercept<\/h3>\n<p id=\"fs-id1165137663560\">Find the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q383732\">Show Solution<\/button><\/p>\n<div id=\"q383732\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137424379\">Set the function equal to zero to solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}0&=\\frac{1}{2}x - 3\\\\[1mm] 3&=\\frac{1}{2}x\\\\[1mm] 6&=x\\\\[1mm] x&=6\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137415633\">The graph crosses the <em>x<\/em>-axis at the point (6,0).<\/p>\n<div id=\"Example_02_02_05\" class=\"example\">\n<div id=\"fs-id1165137805711\" class=\"exercise\">\n<div id=\"fs-id1165135450383\" class=\"commentary\">\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135450388\">A graph of the function is shown in Figure 12. We can see that the <em>x<\/em>-intercept is (6, 0) as we expected.<\/p>\n<p><span id=\"fs-id1165137424274\"> <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010648\/CNX_Precalc_Figure_02_02_0132.jpg\" alt=\"\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727385\" class=\"note precalculus try\">\n<div id=\"ti_02_02_04\" class=\"exercise\">\n<div id=\"fs-id1165134389962\" class=\"problem\">\n<p style=\"text-align: center;\"><strong>Figure 12.\u00a0<\/strong>The graph of the linear function [latex]f\\left(x\\right)=\\frac{1}{2}x - 3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p id=\"fs-id1165134389964\">Find the <em>x<\/em>-intercept of [latex]f\\left(x\\right)=\\frac{1}{4}x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q946091\">Show Solution<\/button><\/p>\n<div id=\"q946091\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(16,\\text{ 0}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<h2 id=\"fs-id1165137410528\">Calculating and Interpreting Slope<\/h2>\n<p id=\"fs-id1165137573268\">In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the <strong>slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex]\u00a0and [latex]{y}_{2}[\/latex] \u2014which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex]\u2014we can calculate the slope [latex]m[\/latex],\u00a0as follows<\/p>\n<div id=\"fs-id1165137757690\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex][\/latex]<\/div>\n<p id=\"fs-id1165137767606\">where [latex]\\Delta y[\/latex] is the vertical displacement and [latex]\\Delta x[\/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] for the function [latex]f[\/latex], [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex], so we could equivalently write<\/p>\n<div id=\"fs-id1165137438737\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex][\/latex]<\/div>\n<p>The graph in Figure 5\u00a0indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex]<br \/>\nand [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex],\u00a0is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010639\/CNX_Precalc_Figure_02_01_005n2.jpg\" alt=\"Graph depicting how to calculate the slope of a line\" width=\"487\" height=\"569\" \/><figcaption class=\"wp-caption-text\"><b>Figure 5<\/b><\/figcaption><\/figure>\n<p><em>The slope of a function is calculated by the change in [latex]y[\/latex] <\/em><\/p>\n<div id=\"Example_02_01_03\" class=\"example\">\n<div id=\"fs-id1165137641892\" class=\"exercise\">\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3>Finding the Slope of a Linear Function<\/h3>\n<p id=\"fs-id1165137724908\">If [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function, and [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q603714\">Show Solution<\/button><\/p>\n<div id=\"q603714\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137714054\">The coordinate pairs are [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]. To find the rate of change, we divide the change in output by the change in input.<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-\\left(-2\\right)}{8 - 3}=\\frac{3}{5}[\/latex]<\/p>\n<p id=\"fs-id1165137469338\">We could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m>0[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137842413\">As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or <em>y<\/em>-coordinate, used corresponds with the first input value, or <em>x<\/em>-coordinate, used.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p id=\"fs-id1165137451238\">If [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function, and [latex]\\left(2,\\text{ }3\\right)[\/latex]\u00a0and [latex]\\left(0,\\text{ }4\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q786191\">Show Solution<\/button><\/p>\n<div id=\"q786191\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]m=\\frac{4 - 3}{0 - 2}=\\frac{1}{-2}=-\\frac{1}{2}[\/latex] ; decreasing because [latex]m<0[\/latex].\n\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm169760\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=169760&theme=lumen&iframe_resize_id=ohm169760&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3>Example 4: Finding the Population Change from a Linear Function<\/h3>\n<p id=\"fs-id1165135386495\">The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q870587\">Show Solution<\/button><\/p>\n<div id=\"q870587\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137415969\">The rate of change relates the change in population to the change in time. The population increased by [latex]27,800 - 23,400=4400[\/latex] people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4,400\\text{ people}}{4\\text{ years}}=1,100\\text{ }\\frac{\\text{people}}{\\text{year}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex][\/latex]<\/p>\n<p id=\"fs-id1165137705484\">So the population increased by 1,100 people per year.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137451439\">Because we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q468046\">Show Solution<\/button><\/p>\n<div id=\"q468046\" class=\"hidden-answer\" style=\"display: none\">[latex]m=\\frac{1,868 - 1,442}{2,012 - 2,009}=\\frac{426}{3}=142\\text{ people per year}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm57147\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=57147&theme=lumen&iframe_resize_id=ohm57147&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":13,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Writing an equation using point slope form given a point and slope\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/FntpEHhLHvw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing an equation using point slope form given two points\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/DhDtKR0VyLE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Write a Slope Intercept Equation for a Line on a Graph\",\"author\":\"\",\"organization\":\"Davitily\",\"url\":\"https:\/\/youtu.be\/L_5tE1vUsyc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Horizontal and Vertical Lines (How to Graph and Write Equations)\",\"author\":\"\",\"organization\":\"Mario\\'s Math Tutoring\",\"url\":\"https:\/\/youtu.be\/G8epj0-fw1Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine if Lines are Parallel, Perpendicular or Neither\",\"author\":\"\",\"organization\":\"Professor Kat\",\"url\":\"https:\/\/youtu.be\/errqXpHe510\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing Equations of Parallel Lines Tutorial\",\"author\":\"\",\"organization\":\"Friendly Math 101\",\"url\":\"https:\/\/youtu.be\/0iQI99ov34I\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing Equations of Perpendicular Lines Tutorial\",\"author\":\"\",\"organization\":\"Friendly Math 101\",\"url\":\"https:\/\/youtu.be\/3ewIVsnHjeA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":61,"module-header":"fresh_take","content_attributions":null,"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/750"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/750\/revisions"}],"predecessor-version":[{"id":5156,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/750\/revisions\/5156"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/750\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=750"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=750"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=750"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=750"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}