2.1 Graphs of linear functions  (Page 18/15)

Two competing telephone companies offer different payment plans. The two plans charge the same rate per long distance minute, but charge a different monthly flat fee. A consumer wants to determine whether the two plans will ever cost the same amount for a given number of long distance minutes used. The total cost of each payment plan can be represented by a linear function. To solve the problem, we will need to compare the functions. In this section, we will consider methods of comparing functions using graphs.

Graphing linear functions

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.

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 $f(x)=x.$

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, $f(x)=2x,$ 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 $\left(1,2\right).$ Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point $\left(2,4\right).$ Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.

Graphing by plotting points

Graph $f(x)=-\frac{2}{3}x+5$ by plotting points.

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.

$$\begin{array}{ccc}x=0& & f(0)=-\frac{2}{3}(0)+5=5\Rightarrow \left(0,5\right)\\ x=3& & f(3)=-\frac{2}{3}(3)+5=3\Rightarrow \left(3,3\right)\\ x=6& & f(6)=-\frac{2}{3}(6)+5=1\Rightarrow \left(6,1\right)\end{array}$$

Plot the coordinate pairs and draw a line through the points. [link] represents the graph of the function $f(x)=-\frac{2}{3}x+5.$