{"id":1092,"date":"2015-11-12T18:35:32","date_gmt":"2015-11-12T18:35:32","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1092"},"modified":"2017-03-31T21:57:36","modified_gmt":"2017-03-31T21:57:36","slug":"graph-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/graph-linear-functions\/","title":{"raw":"Graph linear functions","rendered":"Graph linear functions"},"content":{"raw":"
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In Linear Functions, we saw that that the graph of a linear function is a straight line. We were also able to see the points of the function as well as the initial value from a graph. By graphing two functions, then, we can more easily compare their characteristics.<\/p>\r\n

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. And the third is by using transformations of the identity function [latex]f\\left(x\\right)=x[\/latex].<\/p>\r\n\r\n<\/div>\r\n\u00a0\r\n\r\n\u00a0\r\n

Graphing a Function by Plotting Points<\/span><\/h2>\r\n

To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function, [latex]f\\left(x\\right)=2x[\/latex], we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point (1, 2). Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point (2, 4). Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.<\/p>\r\n\r\n<\/section><\/div>\r\n

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How To: Given a linear function, graph by plotting points.<\/h3>\r\n
  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><\/div>\r\n
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    Example 1: Graphing by Plotting Points<\/h3>\r\n

    Graph [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex] by plotting points.<\/p>\r\n\r\n<\/div>\r\n

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    Solution<\/h3>\r\n

    Begin by choosing input values. This function includes a fraction with a denominator of 3, so let\u2019s choose multiples of 3 as input values. We will choose 0, 3, and 6.<\/p>\r\n

    Evaluate the function at each input value, and use the output value to identify coordinate pairs.<\/p>\r\n\r\n

    [latex]\\begin{cases}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{cases}[\/latex]<\/div>\r\nPlot the coordinate pairs and draw a line through the points. Figure 1 shows\u00a0the graph of the function [latex]f\\left(x\\right)=-\\frac{2}{3}x+5[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"400\"]\"TheFigure 1<\/b>[\/caption]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    Analysis of the Solution<\/h3>\r\n

    The 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.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    Try It 1<\/h3>\r\n

    Graph [latex]f\\left(x\\right)=-\\frac{3}{4}x+6[\/latex] by plotting points.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

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    Graphing a Linear Function Using y-<\/em>intercept and Slope<\/span><\/h2>\r\n

    Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y-<\/em>intercept, which is the point at which the input value is zero. To find the y-<\/em>intercept<\/strong>, we can set x<\/em> = 0 in the equation.<\/p>\r\n

    The other characteristic of the linear function is its slope m<\/em>, which is a measure of its steepness. Recall that the slope is the rate of change of the function. The slope of a function is equal to the ratio of the change in outputs to the change in inputs. Another way to think about the slope is by dividing the vertical difference, or rise, by the horizontal difference, or run. We encountered both the y-<\/em>intercept and the slope in Linear Functions.<\/p>\r\n

    Let\u2019s consider the following function.<\/p>\r\n

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

    The 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 Figure 2.<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"617\"]\"graphFigure 2<\/b>[\/caption]\r\n\r\n
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    A General Note: Graphical Interpretation of a Linear Function<\/h3>\r\n

    In the equation [latex]f\\left(x\\right)=mx+b[\/latex]<\/p>\r\n\r\n