{"id":1734,"date":"2024-06-05T21:39:20","date_gmt":"2024-06-05T21:39:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1734"},"modified":"2025-08-13T23:27:48","modified_gmt":"2025-08-13T23:27:48","slug":"graphs-of-linear-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-linear-functions-learn-it-3\/","title":{"raw":"Graphs of Linear Functions: Learn It 3","rendered":"Graphs of Linear Functions: Learn It 3"},"content":{"raw":"

Graphing Linear Functions<\/h2>\r\nNow that we\u2019ve seen and interpreted graphs of linear functions, let\u2019s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-<\/em>intercept and slope. The third is applying transformations to the identity function [latex]f\\left(x\\right)=x[\/latex].\r\n

Graphing a Function by Plotting Points<\/h3>\r\nTo find points on a function's graph, select input values, evaluate the function at these inputs, and calculate the corresponding outputs. These input-output pairs form coordinates, which you can plot on a grid. To graph the function, you should evaluate it at least two input values to identify at least two points.\r\n\r\n
Given the function [latex]f\\left(x\\right)=2x[\/latex], we might use the input values [latex]1[\/latex] and [latex]2[\/latex].[latex]\\\\[\/latex] Evaluating the function for an input value of [latex]1[\/latex] yields an output value of [latex]2[\/latex] which is represented by the point [latex](1, 2)[\/latex]. [latex]\\\\[\/latex]Evaluating the function for an input value of [latex]2[\/latex] yields an output value of [latex]4[\/latex] which is represented by the point [latex](2, 4)[\/latex].<\/section>
How To: Given a linear function, graph by plotting points.<\/strong>\r\n
    \r\n \t
  1. Choose a minimum of two input values.<\/li>\r\n \t
  2. Evaluate the function at each input value.<\/li>\r\n \t
  3. Use the resulting output values to identify coordinate pairs.<\/li>\r\n \t
  4. Plot the coordinate pairs on a grid.<\/li>\r\n \t
  5. Draw a line through the points.<\/li>\r\n<\/ol>\r\n<\/section>
    Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.<\/section>
    Graph the following by plotting points.\r\n
    [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex]<\/center>\r\n[reveal-answer q=\"589508\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"589508\"]Begin by choosing input values. This function includes a fraction with a denominator of [latex]3[\/latex] so let\u2019s choose multiples of [latex]3[\/latex] as input values. We will choose [latex]0[\/latex], [latex]3[\/latex], and [latex]6[\/latex].\r\n[latex]\\\\[\/latex]\r\nEvaluate the function at each input value and use the output value to identify coordinate pairs.\r\n

    [latex]\\begin{array}{llllll}x=0& & f\\left(0\\right)=-\\frac{2}{3}\\left(0\\right)+5=5\\Rightarrow \\left(0,5\\right)\\\\ x=3& & f\\left(3\\right)=-\\frac{2}{3}\\left(3\\right)+5=3\\Rightarrow \\left(3,3\\right)\\\\ x=6& & f\\left(6\\right)=-\\frac{2}{3}\\left(6\\right)+5=1\\Rightarrow \\left(6,1\\right)\\end{array}[\/latex]<\/p>\r\nPlot the coordinate pairs and draw a line through the points. The graph below is of\u00a0the function [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"400\"]\"The Graph of f(x) with three points labeled[\/caption]Analysis of the Solution<\/strong>\r\n\r\nThe graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant which indicates a negative slope. This is also expected from the negative constant rate of change in the equation for the function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]294294[\/ohm_question]<\/section>\r\n

    Graphing a Linear Function Using [latex]y[\/latex]-intercept and Slope<\/h3>\r\nAnother way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its [latex]y[\/latex]-<\/em>intercept which is the point at which the input value is zero. To find the [latex]y[\/latex]-<\/em>intercept<\/strong>, we can set [latex]x=0[\/latex] in the equation. The other characteristic of the linear function is its slope [latex]m[\/latex].\r\n\r\n
    Keep in mind that if a function has a [latex]y[\/latex]-intercept, we can always find it by setting [latex]x=0[\/latex] and then solving for [latex]y[\/latex].<\/section>
    Let\u2019s consider the following function.\r\n

    [latex]f\\left(x\\right)=\\frac{1}{2}x+1[\/latex]<\/p>\r\n\r\n

      \r\n \t
    • The slope is [latex]\\frac{1}{2}[\/latex]. Because the slope is positive, we know the graph will slant upward from left to right.<\/li>\r\n \t
    • The [latex]y[\/latex]-<\/em>intercept is the point on the graph when [latex]x\u00a0= 0[\/latex]. The graph crosses the [latex]y[\/latex]-axis at [latex](0, 1)[\/latex].<\/li>\r\n<\/ul>\r\nNow we know the slope and the [latex]y[\/latex]-intercept. We can begin graphing by plotting the point [latex](0, 1)[\/latex] We know that the slope is rise over run, [latex]m=\\frac{\\text{rise}}{\\text{run}}[\/latex].\r\n\r\nFrom our example, we have [latex]m=\\frac{1}{2}[\/latex], which means that the rise is [latex]1[\/latex] and the run is [latex]2[\/latex]. Starting from our [latex]y[\/latex]-intercept [latex](0, 1)[\/latex], we can rise [latex]1[\/latex] and then run [latex]2[\/latex] or run [latex]2[\/latex] and then rise [latex]1[\/latex]. We repeat until we have multiple points, and then we draw a line through the points as shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"617\"]\"graph Graph of f(x) with rise and run labeled[\/caption]\r\n\r\n<\/section>
      \r\n

      graphical interpretation of a linear function<\/h3>\r\nIn the equation [latex]f\\left(x\\right)=mx+b[\/latex]\r\n
        \r\n \t
      • [latex]b[\/latex]\u00a0is the [latex]y[\/latex]-intercept of the graph and indicates the point [latex](0, b)[\/latex] at which the graph crosses the [latex]y[\/latex]-axis.<\/li>\r\n \t
      • [latex]m[\/latex]\u00a0is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:<\/li>\r\n<\/ul>\r\n

        [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]<\/p>\r\n\r\n<\/section>

        How To: Given the equation for a linear function, graph the function using the [latex]y[\/latex]-intercept and slope.<\/strong>\r\n
          \r\n \t
        1. Evaluate the function at an input value of zero to find the [latex]y[\/latex]-<\/em>intercept.<\/li>\r\n \t
        2. Identify the slope.<\/li>\r\n \t
        3. Plot the point represented by the y-<\/em>intercept.<\/li>\r\n \t
        4. Use [latex]\\frac{\\text{rise}}{\\text{run}}[\/latex] to determine at least two more points on the line.<\/li>\r\n \t
        5. Draw a line which passes through the points.<\/li>\r\n<\/ol>\r\n<\/section>
          Graph [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex] using the [latex]y[\/latex]-<\/em>intercept and slope.[reveal-answer q=\"507667\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507667\"]Evaluate the function at [latex]x\u00a0= 0[\/latex] to find the [latex]y-[\/latex]intercept. The output value when [latex]x\u00a0= 0[\/latex] is [latex]5[\/latex], so the graph will cross the [latex]y[\/latex]-axis at [latex](0, 5)[\/latex].\r\n[latex]\\\\[\/latex]\r\nAccording to the equation for the function, the slope of the line is [latex]-\\frac{2}{3}[\/latex]. This tells us that for each vertical decrease in the \"rise\" of [latex]\u20132[\/latex] units, the \"run\" increases by [latex]3[\/latex] units in the horizontal direction.\r\n[latex]\\\\[\/latex]\r\nWe can now graph the function by first plotting the [latex]y[\/latex]-intercept. From the initial value [latex](0, 5)[\/latex] we move down [latex]2[\/latex] units and to the right [latex]3[\/latex] units. We can extend the line to the left and right by repeating, and then draw a line through the points.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"graph Graph of f(x)[\/caption]\r\n\r\n[latex]\\\\[\/latex]\r\nAnalysis of the Solution<\/strong>\r\n[latex]\\\\[\/latex]\r\nThe graph slants downward from left to right which means it has a negative slope as expected.[\/hidden-answer]<\/section>","rendered":"

          Graphing Linear Functions<\/h2>\n

          Now that we\u2019ve seen and interpreted graphs of linear functions, let\u2019s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-<\/em>intercept and slope. The third is applying transformations to the identity function [latex]f\\left(x\\right)=x[\/latex].<\/p>\n

          Graphing a Function by Plotting Points<\/h3>\n

          To find points on a function’s graph, select input values, evaluate the function at these inputs, and calculate the corresponding outputs. These input-output pairs form coordinates, which you can plot on a grid. To graph the function, you should evaluate it at least two input values to identify at least two points.<\/p>\n

          Given the function [latex]f\\left(x\\right)=2x[\/latex], we might use the input values [latex]1[\/latex] and [latex]2[\/latex].[latex]\\\\[\/latex] Evaluating the function for an input value of [latex]1[\/latex] yields an output value of [latex]2[\/latex] which is represented by the point [latex](1, 2)[\/latex]. [latex]\\\\[\/latex]Evaluating the function for an input value of [latex]2[\/latex] yields an output value of [latex]4[\/latex] which is represented by the point [latex](2, 4)[\/latex].<\/section>\n
          How To: Given a linear function, graph by plotting points.<\/strong><\/p>\n
            \n
          1. Choose a minimum of two input values.<\/li>\n
          2. Evaluate the function at each input value.<\/li>\n
          3. Use the resulting output values to identify coordinate pairs.<\/li>\n
          4. Plot the coordinate pairs on a grid.<\/li>\n
          5. Draw a line through the points.<\/li>\n<\/ol>\n<\/section>\n
            Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.<\/section>\n
            Graph the following by plotting points.<\/p>\n
            [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex]<\/div>\n