344 CHAPTER 6 Algebra, Graphs, and Functions Slope Another useful concept when you are working with straight lines is slope , which is a measure of the “steepness” of a line. The slope of a line is the ratio of the vertical change, or rise , to the horizontal change, or run , for any two points on the line. Consider Fig. 6.11. Point A has coordinates x y (, ), 1 1 and point B has coordinates x y (, ). 2 2 The vertical change between points A and B is y y . 2 1 − The horizontal change between points A and B is x x . 2 1 − Thus, the slope, which is symbolized with the letter m, can be determined as follows. y x Horizontal change Vertical change y1 y2 (y2 2 y1) x1 x2 A (x1, y1) B (x2, y2) (x2 2 x1) Figure 6.11 Slope of a Line m y x x Slope vertical change horizontal change rise run y 2 1 2 1 = = = − − A line may have a positive slope, a negative slope, or a zero slope, or the slope may be undefined, as indicated in Fig. 6.12. A line with a positive slope rises from left to right, as shown in Fig. 6.12(a). A line with a negative slope falls from left to right, as shown in Fig. 6.12(b). A horizontal line, which neither rises nor falls, has a slope of zero, as shown in Fig. 6.12(c). Since a vertical line does not have any horizontal change (the x-value remains constant) and since we cannot divide by 0, the slope of a vertical line is undefined, as shown in Fig. 6.12(d). y x y x y x y x Positive slope (m . 0) (a) Negative slope (m , 0) (b) Zero slope (m 5 0) (c) Slope is undefined (d) Figure 6.12 Did You Know? Up Up and Away m Ryoyu Kobayashi Although we may not think much about it, the slope of a line is something we are altogether familiar with. You confront it every time you run up the stairs, moving 8 inches horizontally for every 6 inches up. The 2022 Olympic gold medalist Ryoyu Kobayashi of Japan is familiar with the concept of slope. He speeds down a steep 140 meters of ski ramp at speeds of more than 60 mph before he takes flight. Example 5 Determining the Slope of a Line Determine the slope of the line that passes through the points ( 1, 3) − − and (1, 5). Solution Let’s begin by drawing a sketch, illustrating the points and the line. See Fig. 6.13(a). We will let x y (, ) 1 1 be ( 1, 3) − − and x y (, ) 2 2 be (1, 5). Then y y x x Slope 5 ( 3) 1 ( 1) 5 3 1 1 8 2 4 1 4 2 1 2 1 = − − = − − − − = + + = = = The slope of 4 means that there is a vertical change of 4 units for each horizontal change of 1 unit; see Fig. 6.13(b). The slope is positive, and the line rises from left to right. Note that we would have obtained the same result if we let x y (, ) 1 1 be (1, 5) and x y (, ) 2 2 be ( 1, 3). − − Try this method now and see. Marcin Kadziolka/ Shutterstock
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