{"id":911,"date":"2022-02-26T18:15:21","date_gmt":"2022-02-26T18:15:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=911"},"modified":"2025-11-18T21:54:44","modified_gmt":"2025-11-18T21:54:44","slug":"2-2-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-2-linear-functions\/","title":{"raw":"2.2.1: Linear Functions","rendered":"2.2.1: Linear Functions"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Describe rate-initial value form of a linear function<\/li>\r\n \t<li>Describe slope-intercept form of a linear function<\/li>\r\n \t<li>Write the slope-intercept form of a graphed linear function<\/li>\r\n \t<li>Identify the domain and range of a linear function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Rate-Initial Value Form of a Linear Function<\/h2>\r\nEvery linear pattern may be represented or modeled by a linear function. Table 1 shows a linear pattern where\u00a0<span style=\"font-size: 1em;\">[latex]y[\/latex] increases by 3 as [latex]x[\/latex] increases by 1.<\/span><span style=\"font-size: 1em;\">\u00a0<\/span>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tfoot>\r\n<tr>\r\n<td style=\"width: 12.5%;\" colspan=\"8\">Table 1. A linear pattern of [latex]y[\/latex] as a function of [latex]x[\/latex].<\/td>\r\n<\/tr>\r\n<\/tfoot>\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 12.5%;\" colspan=\"8\">Linear Pattern of <em>y<\/em> Increasing by 3 as <em>x<\/em> Increases by 1<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">-3<\/td>\r\n<td style=\"width: 12.5%;\">-2<\/td>\r\n<td style=\"width: 12.5%;\">-1<\/td>\r\n<td style=\"width: 12.5%;\">0<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">2<\/td>\r\n<td style=\"width: 12.5%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex]y[\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">-8<\/td>\r\n<td style=\"width: 12.5%;\">-5<\/td>\r\n<td style=\"width: 12.5%;\">-2<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">4<\/td>\r\n<td style=\"width: 12.5%;\">7<\/td>\r\n<td style=\"width: 12.5%;\">10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"font-size: 1em;\">This means that t<\/span><span style=\"font-size: 1rem; text-align: initial;\">he rate of change of the pattern in table 1 is 3. Therefore, the relationship between [latex]x[\/latex] and [latex]y[\/latex] must include the term [latex]y = 3x[\/latex] to describe the pattern, since [latex]y[\/latex] increases by 3 as [latex]x[\/latex] increases by 1. However, the equation [latex] y = 3x [\/latex] is not quite right since if we plug in 1 for\u00a0[latex]x[\/latex], we get [latex]y=3[\/latex] when [latex]y[\/latex] is in fact equal to 4, according to table 1. To get from 3 to 4 we must add 1, so let's consider the equation [latex]y=3x+1[\/latex] as the representation of the pattern. We can try this new equation out on a couple of data points. When [latex]x=2[\/latex], the equation tells us that [latex]y=3x+1=3(2)+1=7[\/latex], which is correct according to table 1. When [latex]x=-3[\/latex], the equation tells us that [latex]y=3x+1=3(-3)+1=-8[\/latex], which is also correct. We can go on to confirm that each [latex]y[\/latex]-value in table 1 can be found from an [latex]x[\/latex]-value using the equation [latex]y=3x+1[\/latex].<\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">Our first attempt at finding an equation, [latex]y=3x[\/latex] didn't work because the\u00a0value when\u00a0[latex] x = 0 [\/latex] is not 0. The value when [latex] x = 0 [\/latex] is 1. Therefore, we had to add the value of 1 to [latex]3x[\/latex] to get [latex] y = 3x+1[\/latex]. The function form of this equation is\u00a0[latex] f(x) = 3x + 1 [\/latex].<\/span>\r\n<div class=\"textbox shaded\">\r\n<h3>RATE-INITIAL VALUE FORM OF A FUNCTION<\/h3>\r\nEvery linear pattern can be described by a function of the form:\r\n<p style=\"text-align: center;\"><em>dependent variable = (rate of change) \u00b7 (independent variable) + initial value<\/em><\/p>\r\nwhere initial value is defined as the value of the dependent variable when the independent variable equals zero.\r\n\r\n<\/div>\r\nRecall that the rate of change is the change in dependent variable divided by the change in the independent variable. Since this is a lot of verbiage to write every time we are finding a rate of change, we use the Greek capital letter delta, [latex]\\Delta[\/latex], to represent \"change in\". So \"change in [latex]y[\/latex]\" is written [latex]\\Delta y[\/latex] and \"change in [latex]x[\/latex]\" is written [latex]\\Delta x[\/latex]. This greatly simplifies the rate of change to [latex]\\dfrac{\\Delta y}{\\Delta x}[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nWrite an equation for the linear pattern described in the table.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 123.037px;\" colspan=\"6\">Data for a Linear Equation<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.037px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 123.037px;\">0<\/td>\r\n<td style=\"width: 123.037px;\">1<\/td>\r\n<td style=\"width: 123.037px;\">2<\/td>\r\n<td style=\"width: 123.037px;\">3<\/td>\r\n<td style=\"width: 123.066px;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.037px;\">[latex]y[\/latex]<\/td>\r\n<td style=\"width: 123.037px;\">5<\/td>\r\n<td style=\"width: 123.037px;\">3<\/td>\r\n<td style=\"width: 123.037px;\">1<\/td>\r\n<td style=\"width: 123.037px;\">\u20131<\/td>\r\n<td style=\"width: 123.066px;\">\u20133<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nThe initial value at [latex]x=0[\/latex] is 5.\r\n\r\nThe rate of change = [latex]\\frac{\\Delta y}{\\Delta x}=\\frac{-2}{1}=-2[\/latex]\r\n\r\nEquation:\u00a0 \u00a0<em>dependent variable = (rate of change) \u00b7 (independent variable) + initial value<\/em>\r\n<p style=\"text-align: center;\">[latex]y=-2x+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nWrite an equation for the linear pattern described in the table.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 123.037px;\" colspan=\"6\">Data for a Linear Equation<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.037px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 123.037px;\">\u20132<\/td>\r\n<td style=\"width: 123.037px;\">\u20131<\/td>\r\n<td style=\"width: 123.037px;\">0<\/td>\r\n<td style=\"width: 123.037px;\">1<\/td>\r\n<td style=\"width: 123.066px;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.037px;\">[latex]y[\/latex]<\/td>\r\n<td style=\"width: 123.037px;\">\u20138<\/td>\r\n<td style=\"width: 123.037px;\">\u20135<\/td>\r\n<td style=\"width: 123.037px;\">\u20132<\/td>\r\n<td style=\"width: 123.037px;\">1<\/td>\r\n<td style=\"width: 123.066px;\">4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm628\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm628\"]\r\n\r\n[latex]y=3x-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"post-937\" class=\"standard post-937 chapter type-chapter status-publish hentry\">\r\n<h2>Slope-Intercept Form of a Function<\/h2>\r\n<div class=\"entry-content\">\r\n\r\nIn the context of graphing, the constant rate of change is called the slope. For a linear function, the graph is a line which has a constant slope. Earlier we saw that the pattern in table 1 can be represented by the function [latex]f(x)=3x+1[\/latex], where 3 was the rate of change and 1 was the initial value. Since the rate of change was 3, the slope of the line that represents this function, [latex]y=3x+1[\/latex], is also 3. That is, the slope is [latex]\\dfrac{3}{1}=\\dfrac{rise}{run}[\/latex]. The table value [latex] y = 1[\/latex] when [latex]x = 0 [\/latex] is represented by the point (0, 1) on the coordinate plane. This point is the <em><strong>y-intercept<\/strong><\/em> of the line that represents the function and shows the initial value of the function. Therefore, the function of rate-initial value form\r\n<p style=\"text-align: center;\">\u00a0[latex] f(x) = 3x + 1 [\/latex]<\/p>\r\n<p style=\"text-align: left;\">is also called the <em><strong>Slope-Intercept Form<\/strong><\/em> because the rate of change 3 is the slope, and the initial value 1 is the [latex]y[\/latex]-coordinate of the [latex]y[\/latex]-intercept.<\/p>\r\nThe graph of [latex]f(x)=3x+1[\/latex] is shown in figure 1. To illustrate, starting at the [latex]y[\/latex]-intercept (0, 1), the line rises 3 (moving up 3 units) as it runs 1 (moving right 1 unit).\r\n\r\n[caption id=\"attachment_1266\" align=\"aligncenter\" width=\"357\"]<img class=\"wp-image-1266\" style=\"orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/31215404\/y3x11-206x300.png\" alt=\"y = 3x+1\" width=\"357\" height=\"520\" \/> Figure 1. Graph of [latex]f(x)=3x+1[\/latex][\/caption]\r\n<div class=\"textbox shaded\">\r\n<h3>SLOPE-INTERCEPT FORM OF A FUNCTION<\/h3>\r\nEvery line that represents a linear function on the coordinate plane can be written: [latex]y=mx+b[\/latex],\r\n\r\nwhere [latex]m[\/latex] = slope of the line, and [latex]b[\/latex] is the [latex]y[\/latex]-coordinate of the [latex]y[\/latex]-intercept.\r\n\r\nA linear function in slope-intercept form is:\u00a0[latex] f(x) = mx + b[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nThe graph of the function [latex]f(x)=4x-2[\/latex] is a line. In the graphical representation of this function, what is the slope of the line, and what is the [latex]y[\/latex]-intercept?\r\n<h4>Solution<\/h4>\r\n[latex]f(x)=4x-2[\/latex] is graphed as the equation\u00a0[latex]y=4x-2[\/latex]. This is in the form [latex]y=mx+b[\/latex]. So, [latex]m=4[\/latex] and [latex]b=-2[\/latex]. This means that the slope of the line is 4 and the [latex]y[\/latex]-intercept is the point\u00a0 (0, \u20132).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nThe graph of the function [latex]f(x)=-3x+7[\/latex] is a line. In the graphical representation of this function, what is the slope of the line, and what is the [latex]y[\/latex]-intercept?\r\n\r\n[reveal-answer q=\"hjm909\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm909\"]\r\n\r\nSlope = \u20133\r\n\r\n[latex]y[\/latex]-intercept = (0, 7)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nThe graph of a function is a line with a slope of 5 and a [latex]y[\/latex]-intercept at the point (0, \u20136). What function does this represent?\r\n<h4>Solution<\/h4>\r\nThe function is [latex] f(x) = mx + b[\/latex] with [latex]m=5[\/latex] and [latex]b=\u20136[\/latex].\r\n\r\nSo, [latex] f(x) = 5x -6[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nThe graph of a function is a line with a slope of [latex]\\frac{3}{4}[\/latex] and a [latex]y[\/latex]-intercept at the point (0, 3). What function does this represent?\r\n\r\n[reveal-answer q=\"hjm551\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm551\"]\r\n\r\n[latex] f(x) = \\frac{3}{4}x + 3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Slope-Intercept Form on the Coordinate Plane<\/h2>\r\nWe can write the function represented by a line on the coordinate plane as long as the slope and [latex]y[\/latex]-intercept can be identified on the graph.\r\n\r\nThere are two lines graphed in figure 2. The black line has a [latex]y[\/latex]-intercept at the point (0, 3) and a slope of 3 [latex]\\left ( m=\\frac{\\Delta y}{\\Delta x}=\\frac{3-(-9)}{0-(-4)}=\\frac{12}{4}=3 \\right )[\/latex]. Therefore, the function of the black line is\u00a0[latex]f(x) = 3x + 3[\/latex]. The black line has a [latex]y[\/latex]-intercept at (0, \u20135), and its slope is [latex]-\\frac{3}{10}[\/latex]\u00a0[latex]\\left ( m=\\frac{\\Delta y}{\\Delta x}=\\frac{-2-(-5)}{-10-(0)}=\\frac{3}{-10}=-\\frac{3}{10} \\right )[\/latex]. Therefore, the function of the black line is\u00a0[latex]g(x) = -\\frac{3}{10}x -5[\/latex].\r\n\r\n[caption id=\"attachment_1120\" align=\"aligncenter\" width=\"345\"]<img class=\"wp-image-1120\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16015925\/not-quite-perp-300x257.png\" alt=\"Two lines intersecting\" width=\"345\" height=\"294\" \/> Figure 2. Two lines on the coordinate plane.[\/caption]\r\n\r\nBe careful not to confuse the [latex]x[\/latex]-intercept and the\u00a0[latex]y[\/latex]-intercept. The \"<span style=\"color: #0000ff;\">intercept<\/span>\" in the term \"Slope-<span style=\"color: #0000ff;\">intercept<\/span> form\" means the\u00a0[latex]y[\/latex]-intercept. The\u00a0[latex]x[\/latex]-intercept is the intersection point between a graph (e.g., a line) and the [latex]x[\/latex]-axis. Because the intersection point is on the\u00a0[latex]x[\/latex]-axis, its coordinates will be [latex](a, 0)[\/latex]. For the blue line in figure 2, the [latex]x[\/latex]-intercept is the point (-1, 0).\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nDetermine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.\r\n\r\n<img class=\"aligncenter wp-image-1209 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-300x300.png\" alt=\"Line that passes through the points (0, -3) and (2, 0)\" width=\"300\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<h4>Solution<\/h4>\r\nThe [latex]y[\/latex]-intercept is the point where the line intersects the [latex]y[\/latex]-axis: (0, \u20133).\r\n\r\nThe slope can be found between any two points in the graph: (0, \u20133) and (2, 0) lie on the graph, so [latex]m=\\frac{\\Delta y}{\\Delta x}=\\frac{0-(-3)}{2-0}=\\frac{3}{2}[\/latex].\r\n\r\nThe function represented by the line takes the form [latex]f(x)=mx+b[\/latex], so [latex]f(x)=\\frac{3}{2}x-3[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nDetermine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.\r\n\r\n<img class=\"aligncenter wp-image-1210 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-300x300.png\" alt=\"Line that passes through (0, 4) and (2, 0)\" width=\"300\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm269\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm269\"]\r\n\r\n[latex]y[\/latex]-intercept = (0, 4)\r\n\r\nSlope = \u20132\r\n\r\nFunction: [latex]f(x)=-2x+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nDetermine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.\r\n\r\n<img class=\"aligncenter wp-image-1211 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-300x300.png\" alt=\"Horizontal line that passes through (0, 7)\" width=\"300\" height=\"300\" \/>\r\n<h4>Solution<\/h4>\r\nThe line crosses the [latex]y[\/latex]-axis at (0, 7), so the [latex]y[\/latex]-intercept = (0, 7).\r\n\r\nThe line is horizontal so the slope is zero. Using two points on the line (0, 7) and (1, 7) slope = [latex]\\frac{7-7}{1-0}=0[\/latex].\r\n\r\nThe function takes the form [latex]f(x)=mx+b[\/latex] so [latex]f(x)=0x+7[\/latex], which simplifies to [latex]f(x)=7[\/latex].\r\n\r\nNote: all horizontal lines have a slope of zero, so represent a function of the form [latex]f(x)=b[\/latex], where [latex](0, b)[\/latex] is the [latex]y[\/latex]-intercept.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nDetermine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1921\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/29210908\/desmos-graph-97-300x300.png\" alt=\"horizontal line\" width=\"171\" height=\"171\" \/><\/p>\r\n[reveal-answer q=\"hjm542\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm542\"]\r\n\r\n[latex]y[\/latex]-intercept = (0, \u20133)\r\n\r\nSlope = [latex]0[\/latex]\r\n\r\nFunction: [latex]f(x)=-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Domain and Range of a Line<\/h2>\r\nWe have seen that the graph of a linear function is a line on the coordinate plane. For any line that is neither horizontal nor vertical (see figure 3), it extends to the left toward an [latex]x[\/latex]-value of negative infinity and right towards an [latex]x[\/latex]-value of positive infinity. Therefore, its domain, which is the set of all [latex]x[\/latex]-values, is [latex]\\{\\;x\\;|\\;x\\in(-\\infty, \\infty)\\;\\}[\/latex]. It also extends\u00a0<span style=\"font-size: 1em;\">down towards a [latex]y[\/latex]-value of negative infinity and\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">up towards a [latex]y[\/latex]-value of positive infinity. Therefore, its range, which is the set of all [latex]y[\/latex]-values, is\u00a0[latex]\\{\\;f(x)\\;|\\;f(x)\\in(-\\infty, \\infty)\\;\\}[\/latex].<\/span>\r\n\r\n[caption id=\"attachment_1205\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1205 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-300x300.png\" alt=\"A line\" width=\"300\" height=\"300\" \/> Figure 3. Graph of a line.[\/caption]\r\n\r\nFor a <em><strong>horizontal line<\/strong><\/em>, the domain is\u00a0[latex]\\{ \\;x\\;|\\;x\\in (-\\infty, \\infty)\\}[\/latex] because it extends to the left and right on the coordinate plane. However, its range will be a single function-value because all of the points on the horizontal line have the same [latex]y[\/latex]-coordinate. For example, the range of the horizontal line in the figure 4 is [latex]\\{\\;f(x)\\;|\\;f(x)=3\\;\\}[\/latex].\r\n\r\n[caption id=\"attachment_1138\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1138 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-300x300.png\" alt=\"A horizontal line and a vertical line\" width=\"300\" height=\"300\" \/> Figure 4. A horizontal line and a vertical line.[\/caption]\r\n\r\nA <em><strong>vertical line<\/strong><\/em>, like the blue line in figure 4, is NOT a function. Do you see why? [reveal-answer q=\"hjm068\"]Remind me![\/reveal-answer][hidden-answer a=\"hjm068\"]To be a function every [latex]x[\/latex]-value must correspond with exactly one [latex]y[\/latex]-value. In a vertical line the [latex]x[\/latex]-value the line runs up has an infinite number of [latex]y[\/latex]-values. For example, in figure 4, the blue vertical line contains the points (\u20132, 0), (\u20132, 1), (\u20132, 3), (\u20132, \u20135) and (\u20132, any [latex]y[\/latex]-value).[\/hidden-answer]\r\n\r\nHowever, it is a relation and therefore has a domain and a range. Its domain will be a single [latex]x[\/latex]-value because all of the points on the vertical line have the same [latex]x[\/latex]-coordinate. For example, the domain of the vertical line in figure 4 is [latex]\\{ \\;x\\;|\\;x=\u20132\\;\\}[\/latex].\u00a0The range is\u00a0[latex]\\{ \\;y\\;|\\;y\\in (-\\infty, \\infty)\\;\\}[\/latex] because it extends up towards a [latex]y[\/latex]-value of positive infinity and down towards a [latex]y[\/latex]-value of negative infinity.\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nDetermine the domain and range of the function represented by the graph:\r\n\r\n<img class=\"aligncenter wp-image-1214 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-300x298.png\" alt=\"A line\" width=\"300\" height=\"298\" \/>\r\n<h4>Solution<\/h4>\r\nThe line extends to positive and negative infinity in both [latex]x[\/latex] and [latex]y[\/latex] directions, so the domain is [latex]\\{\\;x\\;|\\;x\\in(-\\infty, \\infty)\\}[\/latex] and the range is\u00a0[latex]\\{\\;f(x)\\;|\\;f(x)\\in(-\\infty, \\infty)\\;\\}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nDetermine the domain and range of the function represented by the graph:\r\n\r\n<img class=\"aligncenter wp-image-1215 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-300x222.png\" alt=\"Horizontal line that passes through (0, -6)\" width=\"300\" height=\"222\" \/>\r\n\r\n[reveal-answer q=\"hjm137\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm137\"]\r\n\r\nDomain =[latex]\\{\\;x\\;|\\;x\\in(-\\infty, \\infty)\\}[\/latex]\r\n\r\nRange =[latex]\\{\\;f(x)\\;|\\;f(x)=-6\\;\\}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Describe rate-initial value form of a linear function<\/li>\n<li>Describe slope-intercept form of a linear function<\/li>\n<li>Write the slope-intercept form of a graphed linear function<\/li>\n<li>Identify the domain and range of a linear function<\/li>\n<\/ul>\n<\/div>\n<h2>Rate-Initial Value Form of a Linear Function<\/h2>\n<p>Every linear pattern may be represented or modeled by a linear function. Table 1 shows a linear pattern where\u00a0<span style=\"font-size: 1em;\">[latex]y[\/latex] increases by 3 as [latex]x[\/latex] increases by 1.<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tfoot>\n<tr>\n<td style=\"width: 12.5%;\" colspan=\"8\">Table 1. A linear pattern of [latex]y[\/latex] as a function of [latex]x[\/latex].<\/td>\n<\/tr>\n<\/tfoot>\n<tbody>\n<tr>\n<th style=\"width: 12.5%;\" colspan=\"8\">Linear Pattern of <em>y<\/em> Increasing by 3 as <em>x<\/em> Increases by 1<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 12.5%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 12.5%;\">-3<\/td>\n<td style=\"width: 12.5%;\">-2<\/td>\n<td style=\"width: 12.5%;\">-1<\/td>\n<td style=\"width: 12.5%;\">0<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">2<\/td>\n<td style=\"width: 12.5%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 12.5%;\">[latex]y[\/latex]<\/td>\n<td style=\"width: 12.5%;\">-8<\/td>\n<td style=\"width: 12.5%;\">-5<\/td>\n<td style=\"width: 12.5%;\">-2<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">4<\/td>\n<td style=\"width: 12.5%;\">7<\/td>\n<td style=\"width: 12.5%;\">10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"font-size: 1em;\">This means that t<\/span><span style=\"font-size: 1rem; text-align: initial;\">he rate of change of the pattern in table 1 is 3. Therefore, the relationship between [latex]x[\/latex] and [latex]y[\/latex] must include the term [latex]y = 3x[\/latex] to describe the pattern, since [latex]y[\/latex] increases by 3 as [latex]x[\/latex] increases by 1. However, the equation [latex]y = 3x[\/latex] is not quite right since if we plug in 1 for\u00a0[latex]x[\/latex], we get [latex]y=3[\/latex] when [latex]y[\/latex] is in fact equal to 4, according to table 1. To get from 3 to 4 we must add 1, so let&#8217;s consider the equation [latex]y=3x+1[\/latex] as the representation of the pattern. We can try this new equation out on a couple of data points. When [latex]x=2[\/latex], the equation tells us that [latex]y=3x+1=3(2)+1=7[\/latex], which is correct according to table 1. When [latex]x=-3[\/latex], the equation tells us that [latex]y=3x+1=3(-3)+1=-8[\/latex], which is also correct. We can go on to confirm that each [latex]y[\/latex]-value in table 1 can be found from an [latex]x[\/latex]-value using the equation [latex]y=3x+1[\/latex].<\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Our first attempt at finding an equation, [latex]y=3x[\/latex] didn&#8217;t work because the\u00a0value when\u00a0[latex]x = 0[\/latex] is not 0. The value when [latex]x = 0[\/latex] is 1. Therefore, we had to add the value of 1 to [latex]3x[\/latex] to get [latex]y = 3x+1[\/latex]. The function form of this equation is\u00a0[latex]f(x) = 3x + 1[\/latex].<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>RATE-INITIAL VALUE FORM OF A FUNCTION<\/h3>\n<p>Every linear pattern can be described by a function of the form:<\/p>\n<p style=\"text-align: center;\"><em>dependent variable = (rate of change) \u00b7 (independent variable) + initial value<\/em><\/p>\n<p>where initial value is defined as the value of the dependent variable when the independent variable equals zero.<\/p>\n<\/div>\n<p>Recall that the rate of change is the change in dependent variable divided by the change in the independent variable. Since this is a lot of verbiage to write every time we are finding a rate of change, we use the Greek capital letter delta, [latex]\\Delta[\/latex], to represent &#8220;change in&#8221;. So &#8220;change in [latex]y[\/latex]&#8221; is written [latex]\\Delta y[\/latex] and &#8220;change in [latex]x[\/latex]&#8221; is written [latex]\\Delta x[\/latex]. This greatly simplifies the rate of change to [latex]\\dfrac{\\Delta y}{\\Delta x}[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Write an equation for the linear pattern described in the table.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 123.037px;\" colspan=\"6\">Data for a Linear Equation<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 123.037px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 123.037px;\">0<\/td>\n<td style=\"width: 123.037px;\">1<\/td>\n<td style=\"width: 123.037px;\">2<\/td>\n<td style=\"width: 123.037px;\">3<\/td>\n<td style=\"width: 123.066px;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 123.037px;\">[latex]y[\/latex]<\/td>\n<td style=\"width: 123.037px;\">5<\/td>\n<td style=\"width: 123.037px;\">3<\/td>\n<td style=\"width: 123.037px;\">1<\/td>\n<td style=\"width: 123.037px;\">\u20131<\/td>\n<td style=\"width: 123.066px;\">\u20133<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>The initial value at [latex]x=0[\/latex] is 5.<\/p>\n<p>The rate of change = [latex]\\frac{\\Delta y}{\\Delta x}=\\frac{-2}{1}=-2[\/latex]<\/p>\n<p>Equation:\u00a0 \u00a0<em>dependent variable = (rate of change) \u00b7 (independent variable) + initial value<\/em><\/p>\n<p style=\"text-align: center;\">[latex]y=-2x+5[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Write an equation for the linear pattern described in the table.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 123.037px;\" colspan=\"6\">Data for a Linear Equation<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 123.037px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 123.037px;\">\u20132<\/td>\n<td style=\"width: 123.037px;\">\u20131<\/td>\n<td style=\"width: 123.037px;\">0<\/td>\n<td style=\"width: 123.037px;\">1<\/td>\n<td style=\"width: 123.066px;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 123.037px;\">[latex]y[\/latex]<\/td>\n<td style=\"width: 123.037px;\">\u20138<\/td>\n<td style=\"width: 123.037px;\">\u20135<\/td>\n<td style=\"width: 123.037px;\">\u20132<\/td>\n<td style=\"width: 123.037px;\">1<\/td>\n<td style=\"width: 123.066px;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm628\">Show Answer<\/span><\/p>\n<div id=\"qhjm628\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=3x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"post-937\" class=\"standard post-937 chapter type-chapter status-publish hentry\">\n<h2>Slope-Intercept Form of a Function<\/h2>\n<div class=\"entry-content\">\n<p>In the context of graphing, the constant rate of change is called the slope. For a linear function, the graph is a line which has a constant slope. Earlier we saw that the pattern in table 1 can be represented by the function [latex]f(x)=3x+1[\/latex], where 3 was the rate of change and 1 was the initial value. Since the rate of change was 3, the slope of the line that represents this function, [latex]y=3x+1[\/latex], is also 3. That is, the slope is [latex]\\dfrac{3}{1}=\\dfrac{rise}{run}[\/latex]. The table value [latex]y = 1[\/latex] when [latex]x = 0[\/latex] is represented by the point (0, 1) on the coordinate plane. This point is the <em><strong>y-intercept<\/strong><\/em> of the line that represents the function and shows the initial value of the function. Therefore, the function of rate-initial value form<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]f(x) = 3x + 1[\/latex]<\/p>\n<p style=\"text-align: left;\">is also called the <em><strong>Slope-Intercept Form<\/strong><\/em> because the rate of change 3 is the slope, and the initial value 1 is the [latex]y[\/latex]-coordinate of the [latex]y[\/latex]-intercept.<\/p>\n<p>The graph of [latex]f(x)=3x+1[\/latex] is shown in figure 1. To illustrate, starting at the [latex]y[\/latex]-intercept (0, 1), the line rises 3 (moving up 3 units) as it runs 1 (moving right 1 unit).<\/p>\n<div id=\"attachment_1266\" style=\"width: 367px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1266\" class=\"wp-image-1266\" style=\"orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/31215404\/y3x11-206x300.png\" alt=\"y = 3x+1\" width=\"357\" height=\"520\" \/><\/p>\n<p id=\"caption-attachment-1266\" class=\"wp-caption-text\">Figure 1. Graph of [latex]f(x)=3x+1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>SLOPE-INTERCEPT FORM OF A FUNCTION<\/h3>\n<p>Every line that represents a linear function on the coordinate plane can be written: [latex]y=mx+b[\/latex],<\/p>\n<p>where [latex]m[\/latex] = slope of the line, and [latex]b[\/latex] is the [latex]y[\/latex]-coordinate of the [latex]y[\/latex]-intercept.<\/p>\n<p>A linear function in slope-intercept form is:\u00a0[latex]f(x) = mx + b[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>The graph of the function [latex]f(x)=4x-2[\/latex] is a line. In the graphical representation of this function, what is the slope of the line, and what is the [latex]y[\/latex]-intercept?<\/p>\n<h4>Solution<\/h4>\n<p>[latex]f(x)=4x-2[\/latex] is graphed as the equation\u00a0[latex]y=4x-2[\/latex]. This is in the form [latex]y=mx+b[\/latex]. So, [latex]m=4[\/latex] and [latex]b=-2[\/latex]. This means that the slope of the line is 4 and the [latex]y[\/latex]-intercept is the point\u00a0 (0, \u20132).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>The graph of the function [latex]f(x)=-3x+7[\/latex] is a line. In the graphical representation of this function, what is the slope of the line, and what is the [latex]y[\/latex]-intercept?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm909\">Show Answer<\/span><\/p>\n<div id=\"qhjm909\" class=\"hidden-answer\" style=\"display: none\">\n<p>Slope = \u20133<\/p>\n<p>[latex]y[\/latex]-intercept = (0, 7)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>The graph of a function is a line with a slope of 5 and a [latex]y[\/latex]-intercept at the point (0, \u20136). What function does this represent?<\/p>\n<h4>Solution<\/h4>\n<p>The function is [latex]f(x) = mx + b[\/latex] with [latex]m=5[\/latex] and [latex]b=\u20136[\/latex].<\/p>\n<p>So, [latex]f(x) = 5x -6[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>The graph of a function is a line with a slope of [latex]\\frac{3}{4}[\/latex] and a [latex]y[\/latex]-intercept at the point (0, 3). What function does this represent?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm551\">Show Answer<\/span><\/p>\n<div id=\"qhjm551\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x) = \\frac{3}{4}x + 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Slope-Intercept Form on the Coordinate Plane<\/h2>\n<p>We can write the function represented by a line on the coordinate plane as long as the slope and [latex]y[\/latex]-intercept can be identified on the graph.<\/p>\n<p>There are two lines graphed in figure 2. The black line has a [latex]y[\/latex]-intercept at the point (0, 3) and a slope of 3 [latex]\\left ( m=\\frac{\\Delta y}{\\Delta x}=\\frac{3-(-9)}{0-(-4)}=\\frac{12}{4}=3 \\right )[\/latex]. Therefore, the function of the black line is\u00a0[latex]f(x) = 3x + 3[\/latex]. The black line has a [latex]y[\/latex]-intercept at (0, \u20135), and its slope is [latex]-\\frac{3}{10}[\/latex]\u00a0[latex]\\left ( m=\\frac{\\Delta y}{\\Delta x}=\\frac{-2-(-5)}{-10-(0)}=\\frac{3}{-10}=-\\frac{3}{10} \\right )[\/latex]. Therefore, the function of the black line is\u00a0[latex]g(x) = -\\frac{3}{10}x -5[\/latex].<\/p>\n<div id=\"attachment_1120\" style=\"width: 355px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1120\" class=\"wp-image-1120\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/16015925\/not-quite-perp-300x257.png\" alt=\"Two lines intersecting\" width=\"345\" height=\"294\" \/><\/p>\n<p id=\"caption-attachment-1120\" class=\"wp-caption-text\">Figure 2. Two lines on the coordinate plane.<\/p>\n<\/div>\n<p>Be careful not to confuse the [latex]x[\/latex]-intercept and the\u00a0[latex]y[\/latex]-intercept. The &#8220;<span style=\"color: #0000ff;\">intercept<\/span>&#8221; in the term &#8220;Slope-<span style=\"color: #0000ff;\">intercept<\/span> form&#8221; means the\u00a0[latex]y[\/latex]-intercept. The\u00a0[latex]x[\/latex]-intercept is the intersection point between a graph (e.g., a line) and the [latex]x[\/latex]-axis. Because the intersection point is on the\u00a0[latex]x[\/latex]-axis, its coordinates will be [latex](a, 0)[\/latex]. For the blue line in figure 2, the [latex]x[\/latex]-intercept is the point (-1, 0).<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Determine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1209 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-300x300.png\" alt=\"Line that passes through the points (0, -3) and (2, 0)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/3x-2y62.png 1440w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h4>Solution<\/h4>\n<p>The [latex]y[\/latex]-intercept is the point where the line intersects the [latex]y[\/latex]-axis: (0, \u20133).<\/p>\n<p>The slope can be found between any two points in the graph: (0, \u20133) and (2, 0) lie on the graph, so [latex]m=\\frac{\\Delta y}{\\Delta x}=\\frac{0-(-3)}{2-0}=\\frac{3}{2}[\/latex].<\/p>\n<p>The function represented by the line takes the form [latex]f(x)=mx+b[\/latex], so [latex]f(x)=\\frac{3}{2}x-3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Determine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1210 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-300x300.png\" alt=\"Line that passes through (0, 4) and (2, 0)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-768x771.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-1020x1024.png 1020w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-225x226.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4-350x351.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-2x4.png 1444w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm269\">Show Answer<\/span><\/p>\n<div id=\"qhjm269\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y[\/latex]-intercept = (0, 4)<\/p>\n<p>Slope = \u20132<\/p>\n<p>Function: [latex]f(x)=-2x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Determine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1211 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-300x300.png\" alt=\"Horizontal line that passes through (0, 7)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-768x771.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-1020x1024.png 1020w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-225x226.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7-350x351.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y7.png 1442w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The line crosses the [latex]y[\/latex]-axis at (0, 7), so the [latex]y[\/latex]-intercept = (0, 7).<\/p>\n<p>The line is horizontal so the slope is zero. Using two points on the line (0, 7) and (1, 7) slope = [latex]\\frac{7-7}{1-0}=0[\/latex].<\/p>\n<p>The function takes the form [latex]f(x)=mx+b[\/latex] so [latex]f(x)=0x+7[\/latex], which simplifies to [latex]f(x)=7[\/latex].<\/p>\n<p>Note: all horizontal lines have a slope of zero, so represent a function of the form [latex]f(x)=b[\/latex], where [latex](0, b)[\/latex] is the [latex]y[\/latex]-intercept.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Determine the [latex]y[\/latex]-intercept and slope of the graphed line. Then write the function the line represents.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1921\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/02\/29210908\/desmos-graph-97-300x300.png\" alt=\"horizontal line\" width=\"171\" height=\"171\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm542\">Show Answer<\/span><\/p>\n<div id=\"qhjm542\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y[\/latex]-intercept = (0, \u20133)<\/p>\n<p>Slope = [latex]0[\/latex]<\/p>\n<p>Function: [latex]f(x)=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Domain and Range of a Line<\/h2>\n<p>We have seen that the graph of a linear function is a line on the coordinate plane. For any line that is neither horizontal nor vertical (see figure 3), it extends to the left toward an [latex]x[\/latex]-value of negative infinity and right towards an [latex]x[\/latex]-value of positive infinity. Therefore, its domain, which is the set of all [latex]x[\/latex]-values, is [latex]\\{\\;x\\;|\\;x\\in(-\\infty, \\infty)\\;\\}[\/latex]. It also extends\u00a0<span style=\"font-size: 1em;\">down towards a [latex]y[\/latex]-value of negative infinity and\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">up towards a [latex]y[\/latex]-value of positive infinity. Therefore, its range, which is the set of all [latex]y[\/latex]-values, is\u00a0[latex]\\{\\;f(x)\\;|\\;f(x)\\in(-\\infty, \\infty)\\;\\}[\/latex].<\/span><\/p>\n<div id=\"attachment_1205\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1205\" class=\"wp-image-1205 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-300x300.png\" alt=\"A line\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43-x-2.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1205\" class=\"wp-caption-text\">Figure 3. Graph of a line.<\/p>\n<\/div>\n<p>For a <em><strong>horizontal line<\/strong><\/em>, the domain is\u00a0[latex]\\{ \\;x\\;|\\;x\\in (-\\infty, \\infty)\\}[\/latex] because it extends to the left and right on the coordinate plane. However, its range will be a single function-value because all of the points on the horizontal line have the same [latex]y[\/latex]-coordinate. For example, the range of the horizontal line in the figure 4 is [latex]\\{\\;f(x)\\;|\\;f(x)=3\\;\\}[\/latex].<\/p>\n<div id=\"attachment_1138\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1138\" class=\"wp-image-1138 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-300x300.png\" alt=\"A horizontal line and a vertical line\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/2-2-1-HorVerLine.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1138\" class=\"wp-caption-text\">Figure 4. A horizontal line and a vertical line.<\/p>\n<\/div>\n<p>A <em><strong>vertical line<\/strong><\/em>, like the blue line in figure 4, is NOT a function. Do you see why? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm068\">Remind me!<\/span><\/p>\n<div id=\"qhjm068\" class=\"hidden-answer\" style=\"display: none\">To be a function every [latex]x[\/latex]-value must correspond with exactly one [latex]y[\/latex]-value. In a vertical line the [latex]x[\/latex]-value the line runs up has an infinite number of [latex]y[\/latex]-values. For example, in figure 4, the blue vertical line contains the points (\u20132, 0), (\u20132, 1), (\u20132, 3), (\u20132, \u20135) and (\u20132, any [latex]y[\/latex]-value).<\/div>\n<\/div>\n<p>However, it is a relation and therefore has a domain and a range. Its domain will be a single [latex]x[\/latex]-value because all of the points on the vertical line have the same [latex]x[\/latex]-coordinate. For example, the domain of the vertical line in figure 4 is [latex]\\{ \\;x\\;|\\;x=\u20132\\;\\}[\/latex].\u00a0The range is\u00a0[latex]\\{ \\;y\\;|\\;y\\in (-\\infty, \\infty)\\;\\}[\/latex] because it extends up towards a [latex]y[\/latex]-value of positive infinity and down towards a [latex]y[\/latex]-value of negative infinity.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Determine the domain and range of the function represented by the graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1214 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-300x298.png\" alt=\"A line\" width=\"300\" height=\"298\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-300x298.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-768x764.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-1024x1018.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-225x224.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x-350x348.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y43x.png 1450w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The line extends to positive and negative infinity in both [latex]x[\/latex] and [latex]y[\/latex] directions, so the domain is [latex]\\{\\;x\\;|\\;x\\in(-\\infty, \\infty)\\}[\/latex] and the range is\u00a0[latex]\\{\\;f(x)\\;|\\;f(x)\\in(-\\infty, \\infty)\\;\\}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Determine the domain and range of the function represented by the graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1215 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-300x222.png\" alt=\"Horizontal line that passes through (0, -6)\" width=\"300\" height=\"222\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-300x222.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-768x569.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-1024x758.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-65x48.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-225x167.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6-350x259.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/02\/y-6.png 1510w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm137\">Show Answer<\/span><\/p>\n<div id=\"qhjm137\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain =[latex]\\{\\;x\\;|\\;x\\in(-\\infty, \\infty)\\}[\/latex]<\/p>\n<p>Range =[latex]\\{\\;f(x)\\;|\\;f(x)=-6\\;\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-911\">\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>Linear Functions. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.desmos.com\/calculator\">http:\/\/www.desmos.com\/calculator<\/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":422608,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Linear Functions\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"www.desmos.com\/calculator\",\"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-911","chapter","type-chapter","status-publish","hentry"],"part":963,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/911","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":40,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/911\/revisions"}],"predecessor-version":[{"id":4691,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/911\/revisions\/4691"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/963"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/911\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=911"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=911"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=911"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}