2. What day is it? Lesson: Direct Variation - Constant k??

3. Direct, Joint, and Inverse Variation I can recognize a direct and inverse variation. I can solve direct and inverse variations word problems.

4. Direct Variation up, up As one variable increases, the other must also increase ( up, up ) OR down, down As one variable decreases, the other variable must also decrease. (down, down)

5. Which table represents a direct variation? A. B.

6. Real life? With a shoulder partner take a few minutes to brainstorm real life examples of direct variation. Write them down.With a shoulder partner take a few minutes to brainstorm real life examples of direct variation. Write them down. Food intake/weight Exercise/weight loss Study time/ grades Hourly rate/paycheck size Stress level/blood pressure Recipes Paint Mixtures Drug Manufacturing

7. Direct Variation As one variable increases, the other variable increases. ky=kx x and y are related objects K K is the constant

8. If y varies directly as x, and y=24 and x=3 find: (a) the constant of variation (b) Find y when x=2 (a) Find the constant of variation Write the general equation Substitute

9. (b) Find y when x=2 First we find the constant of variation, which was k=8 Now we substitute into y=kx.

10. Example2: Given a direct variation where y = 12 when x = 3, what is y when x = 10. Step 1: Find k. Step 2: Find the missing variable.

11. Example: Given a direct variation where y = 15 when x = 3, find x when y = 5. Step 1: Find k. Step 2: Find the missing variable.

12. Your TURN Find y when x = 6, if y varies directly as x and y = 8 when x = 2.

13. What does the graph y=kx look like? A linear line with a y-intercept of 0.

14. Looking at the graph, what is the slope of the line? Answer: 3 Looking at the equation, what is the constant of variation? Answer: 3 The constant of variation and the slope are the same!!!!

15. x seconds y frames 1 2 3 4 5 24 48 72 96 120 Ex. As you watch a movie, 24 frames flash by every second. Time (secs.) # of Frames 40 80 100 120 12345 6 0 y = 24x 24 linear equation As x increases, y increases at a constant rate Example of Direct Variation:

16. p varies directly as t. If p = 42 when t = 7, find p when t = 4 Use a proportion to solve: 7p = (42)(4) 7p = 168 p = 24 k = 6 Constant of Variation? y = kx Constant of Variation k or

17. Inverse Variation As one variable increases, the other must decrease ( up, down) OR As one variable decreases, the other variable must increase. (down, up)

18. Inverse Variation Ex Find y when x = 15, if y varies inversely as x and when y = 12, x = 10.

19. Find y when x=15, if y varies inversely as x and x=10 when y=12 Solve by equation:

20. Inverse Variation If y varies inversely as x and x = 18 when y = 6, find y when x = 8. 18(6) = 8y 108 = 8y y = 13.5

21. Your TURN Find x when y = 5, if y varies inversely as x and x = 6 when y = -18. (X 1( 5) = 6(-18)

22. Solve this problem using either method. Find x when y=27, if y varies inversely as x and x=9 when y=45. Answer: 15

23. Inverse Variation Ex. The number of days (x) needed to complete a job varies inversely as the number of workers (y) assigned to a job. If the job can be completed by 2 workers in 30 days. xy = 60 What other combinations of xy also satisfy this relationship? xy 230 320 415 512 610 How many days would it take 3 workers? graph this relationship xy = 60

24. Graphing an Inverse Variation xy = 60 x days y workers 230 320 415 512 610 The graph of an inverse variation relationship is a hyperbola whose center is the origin. Note: as the days double (x 2) the number of workers decreased by its reciprocal, 1/2. 30 20 10 20 300 workersworkers days

25. Graphing an Inverse Variation xy -2-30 -3-20 -4-15 -5-12 -6-10 xy = 60 xy 230 320 415 512 610 not valid for this problem

26. Direct vs. Inverse Variation

27. The cost of hiring a bus for a trip to Niagara Falls is $400. The cost per person (x) varies inversely as the number of persons (y) who will go on the trip. a. find the cost per person if 25 go. b. find the persons who are going if the cost per person is $12.50 xy = k k = $400 x(25) = 400 (cost per person) x (number of persons) = 400 x = 16 a.b.12.50y = 400 y = 32