Section 3.3 Slope of a Line and Rate of Change 219 To move from point A to point B on To move from point A to point B on Park Trail, rise 2 ft and move to the Mt. Dora Trail, rise 5 ft and move right 6 ft (Figure 3-16). to the right 4 ft (Figure 3-17). Mt. Dora Trail Figure 3-16 Figure 3-17 Slope change in y change in x The slope of Mt. Dora Trail is greater than the slope of Park Trail, confirming the observation that Mt. Dora Trail is steeper. On Mt. Dora Trail there is a 5-ft change in elevation for every 4 ft of horizontal distance (a 5:4 ratio). On Park Trail there is only a 2-ft change in elevation for every 6 ft of horizontal distance (a 1:3 ratio). Finding Slope in an Application Example 1 Determine the slope of the ramp up the stairs. Solution: change in y change in x 8 ft 16 ft Write the ratio for the slope and simplify. Slope 8 16 1 2 The slope is 1 2 . Skill Practice 1. Determine the slope of the aircraft’s takeoff path. 5 ft 4 ft 5 4 Slope change in y change in x 2 ft 6 ft 1 3 (Change in x) 6 ft (Change in y) 2 ft Park Trail A B A B (Change in y) 5 ft (Change in x) 4 ft Answer 1. 500 6000 1 12 6000 ft 500 ft 2. Slope Formula The slope of a line may be found using any two points on the line—call these points 1x1, y12 and 1x2, y22. The numbers to the right and below the variables are called subscripts. In this instance, the subscript 1 indicates the coordinates of the first point, and the subscript 2 indicates the coordinates of the second point. The change in y between the points can be found by taking the difference of the y values: The change in x can be found by taking the y2 y1. difference of the x values in the same order: (Figure 3-18). x2 x1 The slope of a line is often symbolized by the letter m and is given by the following formula. 16 ft 8 ft (x2, y2) Figure 3-18 Change in y y2 y1 x2 x1 Change in x (x1, y1) TIP: To find the slope, you can use any two points on the line. The ratio of rise to run will be the same.
miller_beginning_intermediate_algebra_4e_ch1_3
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