34 CHAPTER 1 Graphs Finding and Interpreting the Slope of a Line Given Two Points The slope m of the line containing the points ( ) 1, 2 and ( ) − 5, 3 may be computed as m m 3 2 5 1 5 4 5 4 or as 2 3 1 5 5 4 5 4 ( ) = − − − = − = − = − − − = − = − For every 4-unit change in x, y will change by −5 units. That is, if x increases by 4 units, then y will decrease by 5 units. The average rate of change of y with respect to x is − 5 4 . EXAMPLE 1 Now Work PROBLEMS 13 AND 19 Square Screens To get an undistorted view of slope, the same scale must be used on each axis. However, some graphing utilities have a rectangular screen. Because of this, using the same interval for both x and y will result in a distorted view. For example, Figure 50 shows the graph of the line = y x connecting the points ( ) − − 5, 5 and ( ) 5, 5 . We expect the line to bisect the first and third quadrants, but it doesn’t.We need to adjust the selections for Xmin, Xmax, Ymin, and Ymax so that a square screen results. On a TI-84 Plus CE, this is accomplished by setting the ratio of x to y at 8:5.* Figure 51 shows the graph of the line = y x on a square screen using a TI-84 Plus CE. Notice that the line now bisects the first and third quadrants. Compare this illustration to Figure 50. To get a better idea of the meaning of the slope m of a line, consider the following. Figure 52 = y mx 5 25 Y1 5 0 28 8 Y2 5 x 1 4 Y4 5 x Y5 5 2x Y6 5 6x Y3 5 x 1 2 Exploration On the same square screen, graph the following equations: Y Y x Y x Y x Y x Y x 0 1 4 1 2 2 6 1 2 3 4 5 6 = = = = = = See Figure 52. Slope of the line is 0. Slope of the line is 1 4 . Slope of the line is 1 2 . Slope of the line is 1. Slope of the line is 2. Slope of the line is 6. Figure 53 = y mx 5 25 Y1 5 0 28 8 Y4 5 2x Y5 5 22x Y6 5 26x Y3 5 2 x 1 2 Y2 5 2 x 1 4 Exploration On the same square screen, graph the following equations: = =− =− =− =− =− Y Y x Y x Y x Y x Y x 0 1 4 1 2 2 6 1 2 3 4 5 6 See Figure 53. Slope of the line is 0. Slope of the line is − 1 4 . Slope of the line is − 1 2 . Slope of the line is −1. Slope of the line is −2. Slope of the line is −6. Figure 50 = y x 5 25 (25, 25) (5, 5) 25 5 Figure 51 = y x 5 25 (25, 25) (5, 5) 28 8 *Most graphing utilities have a feature that automatically squares the viewing window. Consult your user’s manual for the appropriate keystrokes.
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