{"id":1902,"date":"2016-06-28T05:53:35","date_gmt":"2016-06-28T05:53:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&p=1902"},"modified":"2018-05-17T00:11:49","modified_gmt":"2018-05-17T00:11:49","slug":"outcome-methods-for-graphing-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/outcome-methods-for-graphing-linear-functions\/","title":{"raw":"Linear Functions and their Graphs","rendered":"Linear Functions and their Graphs"},"content":{"raw":"
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Learning Objectives<\/h3>\r\n
    \r\n \t
  • Define slope for a linear function<\/li>\r\n \t
  • Calculate slope given two points<\/li>\r\n \t
  • Graph a linear function using the slope and y<\/em>-intercept<\/li>\r\n<\/ul>\r\n<\/div>\r\n
    <\/div>\r\n \"An<\/span>\r\n

    Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour.\u00a0In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo species, is a linear function.<\/p>\r\nOne well known\u00a0form for writing linear functions is known as the slope-intercept form<\/strong>, where [latex]x[\/latex] is the input value, [latex]m[\/latex] is the rate of change, and [latex]b[\/latex] is the initial value of the dependant\u00a0variable.\r\n

    [latex]\\begin{array}{cc}\\text{Equation form}\\hfill & y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill & f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\r\nWe often need to calculate the slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex]\u00a0and [latex]{y}_{2}[\/latex] \u2014which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex]\u2014we can calculate the slope [latex]m[\/latex],\u00a0as follows\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\nwhere [latex]\\Delta y[\/latex] is the vertical displacement and [latex]\\Delta x[\/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] for the function [latex]f[\/latex], [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex], so we could equivalently write\r\n

    [latex]m=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\r\nThe graph in Figure 5\u00a0indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex],\u00a0is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.\r\n\r\n\"Graph\r\n\r\nThe slope of a function is calculated by the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex]. It does not matter which coordinate is used as the [latex]\\left({x}_{2,\\text{ }}{y}_{2}\\right)[\/latex] and which is the [latex]\\left({x}_{1},\\text{ }{y}_{1}\\right)[\/latex], as long as each calculation is started with the elements from the same coordinate pair.\r\n\r\nThe units for slope are always [latex]\\frac{\\text{units for the output}}{\\text{units for the input}}[\/latex] Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.\r\n

    \r\n

    Calculate Slope<\/h3>\r\nThe slope, or rate of change, of a function [latex]m[\/latex] can be calculated according to the following:\r\n\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]\r\n\r\nwhere [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex] are input values, [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] are output values.\r\n\r\n<\/div>\r\nWhen the slope of a linear function is positive, the line is moving in an uphill direction across the coordinate axes. This is also called an increasing linear function. Likewise, a decreasing linear function is one whose slope is negative. \u00a0The graph of a decreasing linear function is a line moving in a downhill direction across the coordinate axes.\r\n\r\nIn mathematical terms,\r\n\r\nFor a linear function [latex]f(x)=mx+b[\/latex] if [latex]m>0[\/latex], then [latex]f(x)[\/latex] is an increasing function.\r\n\r\nFor a linear function\u00a0[latex]f(x)=mx+b[\/latex]\u00a0if [latex]m<0[\/latex], then [latex]f(x)[\/latex] is a\u00a0decreasing\u00a0function.\r\n\r\nFor a linear function\u00a0[latex]f(x)=mx+b[\/latex]\u00a0if [latex]m=0[\/latex], then [latex]f(x)[\/latex] is a constant\u00a0function. \u00a0Sometimes we say this is neither increasing nor decreasing.\r\n\r\nIn the following example we will first find the slope of a linear function through two points, then determine whether the line is increasing, decreasing, or neither.\r\n
    \r\n

    Example<\/h3>\r\nIf [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function, and [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?\r\n\r\n[reveal-answer q=\"103343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"103343\"]\r\n\r\nThe coordinate pairs are [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]. To find the rate of change, we divide the change in output by the change in input.\r\n

    [latex]m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-\\left(-2\\right)}{8 - 3}=\\frac{3}{5}[\/latex]<\/p>\r\nWe could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m>0[\/latex].\r\n\r\nAs noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or y<\/em>-coordinate, used corresponds with the first input value, or x<\/em>-coordinate, used.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show examples of how to find the slope of a line passing between two points, then determine whether the line is increasing, decreasing or neither.\r\n\r\nhttps:\/\/youtu.be\/in3NTcx11I8\r\n

    \r\n

    Example<\/h3>\r\nThe population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.\r\n\r\n[reveal-answer q=\"246268\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"246268\"]\r\n\r\nThe rate of change relates the change in population to the change in time. The population increased by [latex]27,800-23,400=4400[\/latex] people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.\r\n

    [latex]\\frac{4,400\\text{ people}}{4\\text{ years}}=1,100\\text{ }\\frac{\\text{people}}{\\text{year}}[\/latex]<\/p>\r\nSo the population increased by 1,100 people per year.\r\n\r\nBecause we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video we show an example where we determine the increase in cost for producing solar panels given two data points.\r\n\r\nhttps:\/\/youtu.be\/4RbniDgEGE4\r\n\r\nThe following video provides na example of how to write a function that will give the cost in dollars for a given number of credit hours taken, x.\r\n\r\nhttps:\/\/youtu.be\/X3Sx2TxH-J0\r\n\r\n<\/div>\r\n

    \u00a0Graph Linear Functions Using Slope and y-Intercept<\/h2>\r\nAnother way to graph a linear function is by using its slope m<\/em>, and y-intercept.\r\n\r\nLet\u2019s consider the following function.\r\n

    [latex]f\\left(x\\right)=\\frac{1}{2}x+1[\/latex]<\/p>\r\nThe slope is [latex]\\frac{1}{2}[\/latex]. Because the slope is positive, we know the graph will slant upward from left to right. The y-<\/em>intercept is the point on the graph when x\u00a0<\/em>= 0. The graph crosses the y<\/em>-axis at (0, 1). Now we know the slope and the y<\/em>-intercept. We can begin graphing by plotting the point (0, 1) We know that the slope is rise over run, [latex]m=\\frac{\\text{rise}}{\\text{run}}[\/latex]. From our example, we have [latex]m=\\frac{1}{2}[\/latex], which means that the rise is 1 and the run is 2. So starting from our y<\/em>-intercept (0, 1), we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in the graph below.\r\n\r\n\"graph\r\n

    \r\n

    A General Note: Graphical Interpretation of a Linear Function<\/h3>\r\nIn the equation [latex]f\\left(x\\right)=mx+b[\/latex]\r\n
      \r\n \t
    • b<\/em>\u00a0is the y<\/em>-intercept of the graph and indicates the point (0, b<\/em>) at which the graph crosses the y<\/em>-axis.<\/li>\r\n \t
    • m<\/em>\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<\/div>\r\nAll linear functions cross the y-axis and therefore have y-intercepts. (Note: A vertical line parallel to the y-axis does not have a y-intercept, but it is not a function.<\/em>)\r\n

      \r\n

      How To: Given the equation for a linear function, graph the function using the y<\/em>-intercept and slope.<\/h3>\r\n
        \r\n \t
      1. Evaluate the function at an input value of zero to find the y-<\/em>intercept.<\/li>\r\n \t
      2. Identify the slope as the rate of change of the input value.<\/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. Sketch the line that passes through the points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Example<\/h3>\r\nGraph [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex] using the y-<\/em>intercept and slope.\r\n\r\n[reveal-answer q=\"421669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421669\"]\r\n\r\nEvaluate the function at [latex]x=0[\/latex]\u00a0to find the y-<\/em>intercept. The output value when [latex]x=0[\/latex] is 5, so the graph will cross the y<\/em>-axis at (0, 5).\r\n\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 3 units in the horizontal direction. We can now graph the function by first plotting the y<\/em>-intercept in the graph below. From the initial value (0, 5) we move down 2 units and to the right 3 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\"graph\r\n\r\nThe graph slants downward from left to right, which means it has a negative slope as expected.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to graph a linear function given the y-intercepts and the slope.\r\n\r\nhttps:\/\/youtu.be\/N6lEPh11gk8\r\n\r\nIn the last example we will show how to graph another linear function using the slope and y-intercept.\r\n
        \r\n

        Example<\/h3>\r\nGraph [latex]f\\left(x\\right)=-\\frac{3}{4}x+6[\/latex]\u00a0using the slope and y-intercept.\r\n[reveal-answer q=\"930515\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"930515\"]\r\n\r\nThe slope of this function is [latex]-\\frac{3}{4}[\/latex] and the y-intercept is [latex](0,6)[\/latex] We can start graphing by plotting the y-intercept and counting down three units and right 4 units. The first stop would be [latex](4,3)[\/latex], and the next stop would be\u00a0[latex](0,8)[\/latex].\r\n

        Answer<\/h4>\r\n\"Screen\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

        Example<\/h3>\r\nThe following figure illustrates the graphs of lines going through the origin [latex](0,0)[\/latex]. Notice, the larger the absolute value of m<\/em>, the steeper the slope.\r\n\r\n\"Graph\r\n\r\nIn the applet below, you can adjust the slope [latex]m[\/latex] and the y-intercept [latex]b[\/latex], and see how that changes the graph of the line [latex]y=mx+b[\/latex], given in slope-intercept form.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/r8ddrnhgwv\r\n\r\n ","rendered":"
        \n
        \n

        Learning Objectives<\/h3>\n
          \n
        • Define slope for a linear function<\/li>\n
        • Calculate slope given two points<\/li>\n
        • Graph a linear function using the slope and y<\/em>-intercept<\/li>\n<\/ul>\n<\/div>\n
          <\/div>\n

          \"An<\/span><\/p>\n

          Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour.\u00a0In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo species, is a linear function.<\/p>\n

          One well known\u00a0form for writing linear functions is known as the slope-intercept form<\/strong>, where [latex]x[\/latex] is the input value, [latex]m[\/latex] is the rate of change, and [latex]b[\/latex] is the initial value of the dependant\u00a0variable.<\/p>\n

          [latex]\\begin{array}{cc}\\text{Equation form}\\hfill & y=mx+b\\hfill \\\\ \\text{Function notation}\\hfill & f\\left(x\\right)=mx+b\\hfill \\end{array}[\/latex]<\/p>\n

          We often need to calculate the slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex]\u00a0and [latex]{y}_{2}[\/latex] \u2014which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex]\u2014we can calculate the slope [latex]m[\/latex],\u00a0as follows<\/p>\n

          [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>\n

          where [latex]\\Delta y[\/latex] is the vertical displacement and [latex]\\Delta x[\/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] for the function [latex]f[\/latex], [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex], so we could equivalently write<\/p>\n

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

          The graph in Figure 5\u00a0indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex]\u00a0and [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex],\u00a0is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.<\/p>\n

          \"Graph<\/p>\n

          The slope of a function is calculated by the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex]. It does not matter which coordinate is used as the [latex]\\left({x}_{2,\\text{ }}{y}_{2}\\right)[\/latex] and which is the [latex]\\left({x}_{1},\\text{ }{y}_{1}\\right)[\/latex], as long as each calculation is started with the elements from the same coordinate pair.<\/p>\n

          The units for slope are always [latex]\\frac{\\text{units for the output}}{\\text{units for the input}}[\/latex] Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.<\/p>\n

          \n

          Calculate Slope<\/h3>\n

          The slope, or rate of change, of a function [latex]m[\/latex] can be calculated according to the following:<\/p>\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>\n

          where [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex] are input values, [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] are output values.<\/p>\n<\/div>\n

          When the slope of a linear function is positive, the line is moving in an uphill direction across the coordinate axes. This is also called an increasing linear function. Likewise, a decreasing linear function is one whose slope is negative. \u00a0The graph of a decreasing linear function is a line moving in a downhill direction across the coordinate axes.<\/p>\n

          In mathematical terms,<\/p>\n

          For a linear function [latex]f(x)=mx+b[\/latex] if [latex]m>0[\/latex], then [latex]f(x)[\/latex] is an increasing function.<\/p>\n

          For a linear function\u00a0[latex]f(x)=mx+b[\/latex]\u00a0if [latex]m<0[\/latex], then [latex]f(x)[\/latex] is a\u00a0decreasing\u00a0function.\n\nFor a linear function\u00a0[latex]f(x)=mx+b[\/latex]\u00a0if [latex]m=0[\/latex], then [latex]f(x)[\/latex] is a constant\u00a0function. \u00a0Sometimes we say this is neither increasing nor decreasing.\n\nIn the following example we will first find the slope of a linear function through two points, then determine whether the line is increasing, decreasing, or neither.\n\n\n

          \n

          Example<\/h3>\n

          If [latex]f\\left(x\\right)[\/latex]\u00a0is a linear function, and [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]\u00a0are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\n

          Show Solution<\/span><\/p>\n
          \n

          The coordinate pairs are [latex]\\left(3,-2\\right)[\/latex]\u00a0and [latex]\\left(8,1\\right)[\/latex]. To find the rate of change, we divide the change in output by the change in input.<\/p>\n

          [latex]m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-\\left(-2\\right)}{8 - 3}=\\frac{3}{5}[\/latex]<\/p>\n

          We could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m>0[\/latex].<\/p>\n

          As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or y<\/em>-coordinate, used corresponds with the first input value, or x<\/em>-coordinate, used.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          In the following video we show examples of how to find the slope of a line passing between two points, then determine whether the line is increasing, decreasing or neither.<\/p>\n