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Graph of the function f(x)= frac <2></p> <p>x plus 1initial value (y-intercept)?

  • one or two points?
  • slope?
  • increasing or decreasing?
  • vertical or horizontal?
    • In this section, you will practice writing liner function equations using the information you've gathered. We will also practice graphing linear functions using different methods and predict how the graphs of linear functions will change when parts of the equation are altered.

    Graph Linear Functions

    We previously 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. 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-intercept and slope. And the third is by using transformations of the identity function [latex]f\left(x\right)=x[/latex].

    Graphing a Function by Plotting Points

    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.

    How To: Given a linear function, graph by plotting points.

    1. Choose a minimum of two input values.
    2. Evaluate the function at each input value.
    3. Use the resulting output values to identify coordinate pairs.
    4. Plot the coordinate pairs on a grid.
    5. Draw a line through the points.

    Example: Graphing by Plotting Points

    Graph [latex]f\left(x\right)=-\frac<2>x+5[/latex] by plotting points.

    Answer: Begin by choosing input values. This function includes a fraction with a denominator of 3, so let’s choose multiples of 3 as input values. We will choose 0, 3, and 6. Evaluate the function at each input value, and use the output value to identify coordinate pairs.

    [latex]\beginx=0& & f\left(0\right)=-\frac\left(0\right)+5=5\Rightarrow \left(0,5\right)\\ x=3& & f\left(3\right)=-\frac\left(3\right)+5=3\Rightarrow \left(3,3\right)\\ x=6& & f\left(6\right)=-\frac\left(6\right)+5=1\Rightarrow \left(6,1\right)\end[/latex]

    The graph of the linear function f\left(x\right)=-\frac</p> <p>Plot the coordinate <a href=pairs and draw a line through the points. The graph below is of the function [latex]f\left(x\right)=-\fracx+5[/latex]. x+5." width="400" height="347" />

    Analysis of the Solution

    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.

    Try It

    Graph [latex]f\left(x\right)=-\frac<3>x+6[/latex] by plotting points.

    cnx_precalc_figure_02_02_0022

    Answer:

    Graphing a Linear Function Using Transformations

    Another option for graphing is to use transformations of the identity function [latex]f\left(x\right)=x[/latex]. A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.

    Vertical Stretch or Compression

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