{"id":731,"date":"2025-07-15T14:54:51","date_gmt":"2025-07-15T14:54:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=731"},"modified":"2026-01-09T20:50:23","modified_gmt":"2026-01-09T20:50:23","slug":"graphs-of-linear-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-linear-functions-learn-it-2\/","title":{"raw":"Graphs of Linear Functions: Learn It 2","rendered":"Graphs of Linear Functions: Learn It 2"},"content":{"raw":"

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 a linear 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].Plot each point on the coordinate plane:\r\n\"\"\r\nThen connect the points with a straight line:\r\n\"\"\r\n\r\n<\/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>
    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\"The\r\n\r\nAnalysis 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>

    If your equation has a fraction, you'll have to divide the number plugged in by whatever is in the denominator. In the previous example, choosing multiples of [latex]3[\/latex] will have the easiest calculation due to the [latex]\\frac{2}{3}[\/latex] in the equation.<\/section>
    [ohm_question hide_question_numbers=1]318705[\/ohm_question]<\/section>","rendered":"

    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 a linear 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].Plot each point on the coordinate plane:
    \n\"\"
    \nThen connect the points with a straight line:
    \n\"\"<\/p>\n<\/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
      Graph the following by plotting points.<\/p>\n
      [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex]<\/div>\n