{"id":836,"date":"2024-04-08T16:44:58","date_gmt":"2024-04-08T16:44:58","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=836"},"modified":"2024-08-21T16:01:38","modified_gmt":"2024-08-21T16:01:38","slug":"basic-functions-and-graphs-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-functions-and-graphs-background-youll-need-2\/","title":{"raw":"Basic Functions and Graphs: Background You\u2019ll Need 2","rendered":"Basic Functions and Graphs: Background You\u2019ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate and interpret the slope of a linear function&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4224,&quot;10&quot;:2,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate and interpret the slope of a linear function<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Linear Functions<\/h2>\r\n<p>Linear functions have the form [latex]f(x)=ax+b[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are constants. In Figure 1, we see examples of linear functions when [latex]a[\/latex] is positive, negative, and zero. Note that if [latex]a&gt;0[\/latex], the graph of the line rises as [latex]x[\/latex] increases. In other words, [latex]f(x)=ax+b[\/latex] is increasing on [latex](\u2212\\infty, \\infty)[\/latex]. If [latex]a&lt;0[\/latex], the graph of the line falls as [latex]x[\/latex] increases. In this case, [latex]f(x)=ax+b[\/latex] is decreasing on [latex](\u2212\\infty, \\infty)[\/latex]. If [latex]a=0[\/latex], the line is horizontal.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202222\/CNX_Calc_Figure_01_02_001.jpg\" alt=\"An image of a graph. The y axis runs from -2 to 5 and the x axis runs from -2 to 5. The graph is of the 3 functions. The first function is \u201cf(x) = 3x + 1\u201d, which is an increasing straight line with an x intercept at ((-1\/3), 0) and a y intercept at (0, 1). The second function is \u201cg(x) = 2\u201d, which is a horizontal line with a y intercept at (0, 2) and no x intercept. The third function is \u201ch(x) = (-1\/2)x\u201d, which is a decreasing straight line with an x intercept and y intercept both at the origin. The function f(x) is increasing at a higher rate than the function h(x) is decreasing.\" width=\"325\" height=\"312\" \/> Figure 1. These linear functions are increasing or decreasing on [latex](-\\infty, \\infty)[\/latex] and one function is a horizontal line.[\/caption]\r\n\r\n<h3>Slope<\/h3>\r\n<p>The graph of any linear function is a line. One of the distinguishing features of a line is its slope. The <strong>slope<\/strong> is the change in [latex]y[\/latex] for each unit change in [latex]x[\/latex]. The slope measures both the steepness and the direction of a line.<\/p>\r\n<p>To calculate the slope of a line, we need to determine the ratio of the change in [latex]y[\/latex] versus the change in [latex]x[\/latex]. To do so, we choose any two points [latex](x_1,y_1)[\/latex] and [latex](x_2,y_2)[\/latex] on the line and calculate [latex]\\dfrac{y_2-y_1}{x_2-x_1}[\/latex]. In Figure 2, we see this ratio is independent of the points chosen.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"465\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202225\/CNX_Calc_Figure_01_02_021.jpg\" alt=\"An image of a graph. The y axis runs from -1 to 10 and the x axis runs from -1 to 6. The graph is of a function that is an increasing straight line. There are four points labeled on the function at (1, 1), (2, 3), (3, 5), and (5, 9). There is a dotted horizontal line from the labeled function point (1, 1) to the unlabeled point (3, 1) which is not on the function, and then dotted vertical line from the unlabeled point (3, 1), which is not on the function, to the labeled function point (3, 5). These two dotted have the label \u201c(y2 - y1)\/(x2 - x1) = (5 -1)\/(3 - 1) = 2\u201d. There is a dotted horizontal line from the labeled function point (2, 3) to the unlabeled point (5, 3) which is not on the function, and then dotted vertical line from the unlabeled point (5, 3), which is not on the function, to the labeled function point (5, 9). These two dotted have the label \u201c(y2 - y1)\/(x2 - x1) = (9 -3)\/(5 - 2) = 2\u201d.\" width=\"465\" height=\"459\" \/> Figure 2. or any linear function, the slope [latex](y_2-y_1)\/(x_2-x_1)[\/latex] is independent of the choice of points [latex](x_1,y_1)[\/latex] and [latex](x_2,y_2)[\/latex] on the line.[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>slope of a linear line<\/h3>\r\n\r\nConsider line [latex]L[\/latex] passing through points [latex](x_1,y_1)[\/latex] and [latex](x_2,y_2)[\/latex]. Let [latex]\\Delta y=y_2-y_1[\/latex] and [latex]\\Delta x=x_2-x_1[\/latex] denote the changes in [latex]y[\/latex] and [latex]x[\/latex], respectively. The slope of the line is<center>[latex]m=\\dfrac{y_2-y_1}{x_2-x_1}=\\dfrac{\\Delta y}{\\Delta x}[\/latex]<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal.<\/p>\r\n<\/section>\r\n<h3>Slope-Intercept Form<\/h3>\r\n<p>The linear equation [latex]f(x)=ax+b[\/latex] encapsulates two crucial pieces of information about its graph: the slope and the [latex]y[\/latex]-intercept. The coefficient '[latex]a[\/latex]' is the slope, dictating the angle and direction of the line, while '[latex]b[\/latex]' gives us the [latex]y[\/latex]-intercept, the point where the line crosses the [latex]y[\/latex]-axis. This equation is the essence of the <strong>slope-intercept form<\/strong>, commonly written as [latex]f(x)=mx+b[\/latex], with '[latex]m[\/latex]' signifying the slope. It succinctly represents the linear function, offering a clear view of its gradient and starting point on a graph.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>slope-intercept form<\/h3>\r\n<p id=\"fs-id1170573425678\">Consider a line with slope [latex]m[\/latex] and [latex]y[\/latex]-intercept [latex](0,b)[\/latex]. The equation<\/p>\r\n<div id=\"fs-id1170573240220\" class=\"equation\" style=\"text-align: center;\">[latex]y=mx+b[\/latex]<\/div>\r\n<div>\u00a0<\/div>\r\n<p id=\"fs-id1170573336985\">is an equation for that line in <strong>slope-intercept form<\/strong>.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573402450\">Consider the line passing through the points [latex](11,-4)[\/latex] and [latex](-4,5)[\/latex], as shown in Figure 3.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"694\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202228\/CNX_Calc_Figure_01_02_002.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 12 and the y axis runs from -5 to 6. The graph is of the function that is a decreasing straight line. The function has two points plotted, at (-4, 5) and (11, 4).\" width=\"694\" height=\"459\" \/> Figure 3. Finding the equation of a linear function with a graph that is a line between two given points.[\/caption]\r\n\r\n<div class=\"wp-caption-text\">\u00a0<\/div>\r\n<ol id=\"fs-id1170573248716\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Find the slope of the line.<\/li>\r\n\t<li>Find an equation for this linear function in slope-intercept form.<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170573411739\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573411739\"]<\/p>\r\n<ol id=\"fs-id1170573411739\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>The slope of the line is<br \/>\r\n<div id=\"fs-id1170573334269\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\dfrac{y_2-y_1}{x_2-x_1}=\\dfrac{5-(-4)}{-4-11}=-\\dfrac{9}{15}=-\\dfrac{3}{5}[\/latex].<br \/>\r\n<br \/>\r\n<\/div>\r\n<\/li>\r\n\t<li>To find the equation in slope-intercept form, we need the slope and the [latex]y[\/latex]-intercept of the line. <br \/>\r\n<br \/>\r\nSince we know the slope to be [latex]-\\dfrac{3}{5}[\/latex],we need to calculate the y-intercept using the slope and one of the points. <br \/>\r\n<br \/>\r\nUsing the point [latex](11, -4)[\/latex], with the line equation [latex] y=mx+b[\/latex]. We have:<br \/>\r\n<center>[latex]\\begin{array}{rl} &amp; -4 = \\frac{3}{5}(11) + b \\\\ &amp; -4 = \\frac{33}{5} + b \\\\ \\text{To find } b, \\text{ we add } \\frac{33}{5} \\text{ to both sides:} &amp; \\\\ &amp; b = -4 + \\frac{33}{5} \\\\ &amp; b = \\frac{-20}{5} + \\frac{33}{5} \\\\ &amp; b = \\frac{13}{5} \\end{array} [\/latex]<\/center>Hence, the equation of the line in slope-intercept form is:\r\n\r\n<div id=\"fs-id1170573274197\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=-\\frac{3}{5}x+\\frac{13}{5}[\/latex].<\/div>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573413936\">Aisha leaves her house at 5:50 a.m. and goes for a [latex]9[\/latex]-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Aisha runs at a constant pace.<\/p>\r\n<ol id=\"fs-id1170573534412\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Describe the distance [latex]D[\/latex] (in miles) Aisha runs as a linear function of her run time [latex]t[\/latex] (in minutes).<\/li>\r\n\t<li>Sketch a graph of [latex]D[\/latex].<\/li>\r\n\t<li>Interpret the meaning of the slope.<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170573413689\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573413689\"]<\/p>\r\n<ol id=\"fs-id1170573413689\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>At time [latex]t=0[\/latex], Aisha is at her house, so [latex]D(0)=0[\/latex]. At time [latex]t=78[\/latex] minutes, Aisha has finished running [latex]9[\/latex] mi, so [latex]D(78)=9[\/latex]. The slope of the linear function is<br \/>\r\n<div id=\"fs-id1170573404013\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\dfrac{9-0}{78-0}=\\dfrac{3}{26}[\/latex]<\/div>\r\n<p>The [latex]y[\/latex]-intercept is [latex](0,0)[\/latex], so the equation for this linear function is<\/p>\r\n<div id=\"fs-id1170573573845\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]D(t)=\\frac{3}{26}t[\/latex]<\/div>\r\n<\/li>\r\n\t<li>To graph [latex]D[\/latex], use the fact that the graph passes through the origin and has slope [latex]m=\\frac{3}{26}[\/latex].\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202230\/CNX_Calc_Figure_01_02_003.jpg\" alt=\"An image of a graph. The y axis is labeled \u201cy, distance in miles\u201d. The x axis is labeled \u201ct, time in minutes\u201d. The graph is of the function \u201cD(t) = 3t\/26\u201d, which is an increasing straight line that starts at the origin. The function ends at the plotted point (78, 9).\" width=\"731\" height=\"190\" \/> Figure 4. Graph of function [latex]D[\/latex] \u2013\u00a0Aisha 's distance from home in miles vs. minutes spent running.[\/caption]\r\n<\/li>\r\n\t<li>The slope [latex]m=\\dfrac{3}{26} \\approx 0.115[\/latex] describes the distance (in miles) Aisha runs per minute, or her average velocity.<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288195[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate and interpret the slope of a linear function&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4224,&quot;10&quot;:2,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate and interpret the slope of a linear function<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Linear Functions<\/h2>\n<p>Linear functions have the form [latex]f(x)=ax+b[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are constants. In Figure 1, we see examples of linear functions when [latex]a[\/latex] is positive, negative, and zero. Note that if [latex]a>0[\/latex], the graph of the line rises as [latex]x[\/latex] increases. In other words, [latex]f(x)=ax+b[\/latex] is increasing on [latex](\u2212\\infty, \\infty)[\/latex]. If [latex]a<0[\/latex], the graph of the line falls as [latex]x[\/latex] increases. In this case, [latex]f(x)=ax+b[\/latex] is decreasing on [latex](\u2212\\infty, \\infty)[\/latex]. If [latex]a=0[\/latex], the line is horizontal.<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202222\/CNX_Calc_Figure_01_02_001.jpg\" alt=\"An image of a graph. The y axis runs from -2 to 5 and the x axis runs from -2 to 5. The graph is of the 3 functions. The first function is \u201cf(x) = 3x + 1\u201d, which is an increasing straight line with an x intercept at ((-1\/3), 0) and a y intercept at (0, 1). The second function is \u201cg(x) = 2\u201d, which is a horizontal line with a y intercept at (0, 2) and no x intercept. The third function is \u201ch(x) = (-1\/2)x\u201d, which is a decreasing straight line with an x intercept and y intercept both at the origin. The function f(x) is increasing at a higher rate than the function h(x) is decreasing.\" width=\"325\" height=\"312\" \/><figcaption class=\"wp-caption-text\">Figure 1. These linear functions are increasing or decreasing on [latex](-\\infty, \\infty)[\/latex] and one function is a horizontal line.<\/figcaption><\/figure>\n<h3>Slope<\/h3>\n<p>The graph of any linear function is a line. One of the distinguishing features of a line is its slope. The <strong>slope<\/strong> is the change in [latex]y[\/latex] for each unit change in [latex]x[\/latex]. The slope measures both the steepness and the direction of a line.<\/p>\n<p>To calculate the slope of a line, we need to determine the ratio of the change in [latex]y[\/latex] versus the change in [latex]x[\/latex]. To do so, we choose any two points [latex](x_1,y_1)[\/latex] and [latex](x_2,y_2)[\/latex] on the line and calculate [latex]\\dfrac{y_2-y_1}{x_2-x_1}[\/latex]. In Figure 2, we see this ratio is independent of the points chosen.<\/p>\n<figure style=\"width: 465px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202225\/CNX_Calc_Figure_01_02_021.jpg\" alt=\"An image of a graph. The y axis runs from -1 to 10 and the x axis runs from -1 to 6. The graph is of a function that is an increasing straight line. There are four points labeled on the function at (1, 1), (2, 3), (3, 5), and (5, 9). There is a dotted horizontal line from the labeled function point (1, 1) to the unlabeled point (3, 1) which is not on the function, and then dotted vertical line from the unlabeled point (3, 1), which is not on the function, to the labeled function point (3, 5). These two dotted have the label \u201c(y2 - y1)\/(x2 - x1) = (5 -1)\/(3 - 1) = 2\u201d. There is a dotted horizontal line from the labeled function point (2, 3) to the unlabeled point (5, 3) which is not on the function, and then dotted vertical line from the unlabeled point (5, 3), which is not on the function, to the labeled function point (5, 9). These two dotted have the label \u201c(y2 - y1)\/(x2 - x1) = (9 -3)\/(5 - 2) = 2\u201d.\" width=\"465\" height=\"459\" \/><figcaption class=\"wp-caption-text\">Figure 2. or any linear function, the slope [latex](y_2-y_1)\/(x_2-x_1)[\/latex] is independent of the choice of points [latex](x_1,y_1)[\/latex] and [latex](x_2,y_2)[\/latex] on the line.<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>slope of a linear line<\/h3>\n<p>Consider line [latex]L[\/latex] passing through points [latex](x_1,y_1)[\/latex] and [latex](x_2,y_2)[\/latex]. Let [latex]\\Delta y=y_2-y_1[\/latex] and [latex]\\Delta x=x_2-x_1[\/latex] denote the changes in [latex]y[\/latex] and [latex]x[\/latex], respectively. The slope of the line is<\/p>\n<div style=\"text-align: center;\">[latex]m=\\dfrac{y_2-y_1}{x_2-x_1}=\\dfrac{\\Delta y}{\\Delta x}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal.<\/p>\n<\/section>\n<h3>Slope-Intercept Form<\/h3>\n<p>The linear equation [latex]f(x)=ax+b[\/latex] encapsulates two crucial pieces of information about its graph: the slope and the [latex]y[\/latex]-intercept. The coefficient &#8216;[latex]a[\/latex]&#8216; is the slope, dictating the angle and direction of the line, while &#8216;[latex]b[\/latex]&#8216; gives us the [latex]y[\/latex]-intercept, the point where the line crosses the [latex]y[\/latex]-axis. This equation is the essence of the <strong>slope-intercept form<\/strong>, commonly written as [latex]f(x)=mx+b[\/latex], with &#8216;[latex]m[\/latex]&#8216; signifying the slope. It succinctly represents the linear function, offering a clear view of its gradient and starting point on a graph.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>slope-intercept form<\/h3>\n<p id=\"fs-id1170573425678\">Consider a line with slope [latex]m[\/latex] and [latex]y[\/latex]-intercept [latex](0,b)[\/latex]. The equation<\/p>\n<div id=\"fs-id1170573240220\" class=\"equation\" style=\"text-align: center;\">[latex]y=mx+b[\/latex]<\/div>\n<div>\u00a0<\/div>\n<p id=\"fs-id1170573336985\">is an equation for that line in <strong>slope-intercept form<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573402450\">Consider the line passing through the points [latex](11,-4)[\/latex] and [latex](-4,5)[\/latex], as shown in Figure 3.<\/p>\n<figure style=\"width: 694px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202228\/CNX_Calc_Figure_01_02_002.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 12 and the y axis runs from -5 to 6. The graph is of the function that is a decreasing straight line. The function has two points plotted, at (-4, 5) and (11, 4).\" width=\"694\" height=\"459\" \/><figcaption class=\"wp-caption-text\">Figure 3. Finding the equation of a linear function with a graph that is a line between two given points.<\/figcaption><\/figure>\n<div class=\"wp-caption-text\">\u00a0<\/div>\n<ol id=\"fs-id1170573248716\" style=\"list-style-type: lower-alpha;\">\n<li>Find the slope of the line.<\/li>\n<li>Find an equation for this linear function in slope-intercept form.<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573411739\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573411739\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170573411739\" style=\"list-style-type: lower-alpha;\">\n<li>The slope of the line is\n<div id=\"fs-id1170573334269\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\dfrac{y_2-y_1}{x_2-x_1}=\\dfrac{5-(-4)}{-4-11}=-\\dfrac{9}{15}=-\\dfrac{3}{5}[\/latex].<\/p>\n<\/div>\n<\/li>\n<li>To find the equation in slope-intercept form, we need the slope and the [latex]y[\/latex]-intercept of the line.\n<p>Since we know the slope to be [latex]-\\dfrac{3}{5}[\/latex],we need to calculate the y-intercept using the slope and one of the points. <\/p>\n<p>Using the point [latex](11, -4)[\/latex], with the line equation [latex]y=mx+b[\/latex]. We have:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rl} & -4 = \\frac{3}{5}(11) + b \\\\ & -4 = \\frac{33}{5} + b \\\\ \\text{To find } b, \\text{ we add } \\frac{33}{5} \\text{ to both sides:} & \\\\ & b = -4 + \\frac{33}{5} \\\\ & b = \\frac{-20}{5} + \\frac{33}{5} \\\\ & b = \\frac{13}{5} \\end{array}[\/latex]<\/div>\n<p>Hence, the equation of the line in slope-intercept form is:<\/p>\n<div id=\"fs-id1170573274197\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=-\\frac{3}{5}x+\\frac{13}{5}[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573413936\">Aisha leaves her house at 5:50 a.m. and goes for a [latex]9[\/latex]-mile run. She returns to her house at 7:08 a.m. Answer the following questions, assuming Aisha runs at a constant pace.<\/p>\n<ol id=\"fs-id1170573534412\" style=\"list-style-type: lower-alpha;\">\n<li>Describe the distance [latex]D[\/latex] (in miles) Aisha runs as a linear function of her run time [latex]t[\/latex] (in minutes).<\/li>\n<li>Sketch a graph of [latex]D[\/latex].<\/li>\n<li>Interpret the meaning of the slope.<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573413689\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573413689\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170573413689\" style=\"list-style-type: lower-alpha;\">\n<li>At time [latex]t=0[\/latex], Aisha is at her house, so [latex]D(0)=0[\/latex]. At time [latex]t=78[\/latex] minutes, Aisha has finished running [latex]9[\/latex] mi, so [latex]D(78)=9[\/latex]. The slope of the linear function is\n<div id=\"fs-id1170573404013\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m=\\dfrac{9-0}{78-0}=\\dfrac{3}{26}[\/latex]<\/div>\n<p>The [latex]y[\/latex]-intercept is [latex](0,0)[\/latex], so the equation for this linear function is<\/p>\n<div id=\"fs-id1170573573845\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]D(t)=\\frac{3}{26}t[\/latex]<\/div>\n<\/li>\n<li>To graph [latex]D[\/latex], use the fact that the graph passes through the origin and has slope [latex]m=\\frac{3}{26}[\/latex].<br \/>\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202230\/CNX_Calc_Figure_01_02_003.jpg\" alt=\"An image of a graph. The y axis is labeled \u201cy, distance in miles\u201d. The x axis is labeled \u201ct, time in minutes\u201d. The graph is of the function \u201cD(t) = 3t\/26\u201d, which is an increasing straight line that starts at the origin. The function ends at the plotted point (78, 9).\" width=\"731\" height=\"190\" \/><figcaption class=\"wp-caption-text\">Figure 4. Graph of function [latex]D[\/latex] \u2013\u00a0Aisha &#8216;s distance from home in miles vs. minutes spent running.<\/figcaption><\/figure>\n<\/li>\n<li>The slope [latex]m=\\dfrac{3}{26} \\approx 0.115[\/latex] describes the distance (in miles) Aisha runs per minute, or her average velocity.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288195\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288195&theme=lumen&iframe_resize_id=ohm288195&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"background_you_need","content_attributions":[],"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/836"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":25,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/836\/revisions"}],"predecessor-version":[{"id":4855,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/836\/revisions\/4855"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/836\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=836"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=836"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=836"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=836"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}