1. Direct Variation (3-4) Objective: Write and graph direct variation equations. Solve problems involving direct variation.

2. Direct Variation Equations A direct variation is described by an equation of the form y = kx, where k ≠ 0. The equation y = kx illustrates a constant rate of change and k is the constant of variation, also called the constant of proportionality.

3. Example 1 Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. a.. y = 2x y = kx k = 2 m = 2

4. Example 1 Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. b.. y = -4x y = kx k = -4 m = -4

5. Direct Variation Equations The slope of the graph of y = kx is k. Since 0 = k(0), the graph of y = kx always passes through the origin. Therefore the x- and y-intercepts are zero.

6. Example 2 Graph y = –The first point is at the origin.

7. Direct Variation Graphs The graphs of all direct variation equations share some common characteristics. Direct variation equations are of the form y = kx, where k ≠ 0. The graph of y = kx always passes through the origin. The slope is positive if k > 0 and the line will rise from left to right. The slope is negative if k < 0 and the line will fall from left to right.

8. Direct Variation Relationships If the relationship between the values of y and x can be described by a direct variation equation, then we say that y varies directly as x.

9. Example 3 Suppose y varies directly as x, and y = 9 when x = -3. a.Write a direct variation equation that relates x and y. -3

10. Example 3 Suppose y varies directly as x, and y = 9 when x = -3. b.Use the direct variation equation to find x when y = 15. -3

11. Direct Variation Problems 1. One of the most common applications of direct variation is the formula d = rt. Distance d varies directly as time t, and the rate r is the constant of variation.

12. Example 4 1. The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. a.Write a direct variation equation to find the distance d driven in time t. 5.5

13. Example 4 1. The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. b.Graph the equation.

14. Example 4 1. The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours. c.Estimate how many hours it would take to drive 500 miles. 60

15. Check Your Progress Choose the best answer for the following. A.Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. A.Constant of variation: 4; slope: -4 B.Constant of variation: 4; slope: 4 C.Constant of variation: -4; slope: -4 D.Constant of variation: ¼; slope: ¼

16. Check Your Progress Choose the best answer for the following. B.Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. A.Constant of variation: 3; slope: 3 B.Constant of variation: 1 / 3 ; slope: 1 / 3 C.Constant of variation: 0; slope: 0 D.Constant of variation: -3; slope: -3

17. Check Your Progress Choose the best answer for the following. –Graph y = 2x. A. B. C. D.

18. Check Your Progress Choose the best answer for the following. A.Suppose y varies directly as x, and y = 15 when x = 5. Write a direct variation equation that relates x and y. A.y = 3x B.y = 15x C.y = 5x D.y = 45x 5

19. Check Your Progress Choose the best answer for the following. B.Suppose y varies directly as x, and y = 15 when x = 5. Use the direct variation equation to find x when y = -45. A.-3 B.9 C.-15 D.-5 3

20. Check Your Progress Choose the best answer for the following. A.Dustin ran a 26-mile marathon in 3.25 hours. Write a direct variation equation to find the distance run for any number of hours. A.d = h B.d = 8h C.d = 8 D.d = 1 / 8 h 3.25

21. Check Your Progress Choose the best answer for the following. B.Dustin ran a 26-mile marathon in 3.25 hours. Graph the equation. Distance (miles) Time (hours) Distance (miles) Time (hours) Distance (miles) Time (hours) Distance (miles) Time (hours) A. B. C. D.

22. Check Your Progress Choose the best answer for the following. C.Estimate how many hours it would take Dustin to run 16 miles. A.1 hour B.½ hour C.16 hours D.2 hours 8