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Warm up

Warm up. Determine the asymptotes for:. Lesson 3-8 Direct, Inverse & Joint Variation. Objective: To recognize and use direct variation to solve problems. Definition: Y varies directly as x means that y = kx where k is the constant of variation. Another way of writing this is k =.

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Warm up

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  1. Warm up • Determine the asymptotes for:

  2. Lesson 3-8 Direct, Inverse & Joint Variation Objective: To recognize and use direct variation to solve problems

  3. Definition: Y varies directly as x means that y = kx where k is the constant of variation. Another way of writing this is k = In other words: * As x increases in value, y increases or * As x decreases in value, y decreases.

  4. Examples of Direct Variation: Note: X decreases, 30, 15, 9 And Y decreases. 10, 5, 3 What is the constant of variation of the table above? Since y = kx we can say Therefore: 10/30=k or k = 1/3 5/15=k or k = 1/3 3/9=k or k =1/3 Note k stays constant. y = 1/3x is the equation!

  5. Direct Variation Direct variation uses the following formula:

  6. Direct Variation example: if y varies directly as x and y = 10 as x = 2.4, find x when y =15. what x and y go together?

  7. Direct Variation if y varies directly as x and y = 10 as x = 2.4, find x when y =15

  8. Direct Variation Example: If y varies directly as the square of x and y = 30 when x = 4, find x when y=270. y=kx2 30=k42 k=1.875 270=1.875x2 x=12

  9. Inverse Variation Inverse is very similar to direct, but in an inverse relationship as one value goesup, the other goes down.

  10. Inverse Variation If y varies inversely as x, then for some constant k. k is still called the constant of variation.

  11. Inverse Variation • With Direct variation we Divide our x’s and y’s. • In Inverse variation we will Multiply them. • x1y1 = x2y2

  12. Inverse Variation If y varies inversely with x and y = 12 when x = 2, find y when x = 8. x1y1 = x2y2 2(12) = 8y 24 = 8y y = 3

  13. 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

  14. Practice • If t varies inversely as q. If t = 240 when q = 0.01, then find the value of t when q = 8 • Two rectangles have the same area. The length of a rectangle varies inversely as the width. If the length of a rectangle is 20 ft when the width is 8 ft, find the length of the rectangle when the width is 10 ft.?

  15. Joint and Combined Variation • Joint variation is like direct variation but it involves more than one quantity. • Combined variation combines both direct and inverse variation in the same problem.

  16. Joint and Combined Variation • For example: if z varies jointly with x & y, then z=kxy. • Ex: if y varies inversely with the square of x, then y=k/x2. • Ex: if z varies directly with y and inversely with x, then z=ky/x.

  17. Example • y varies jointly as x and w and inversely as the square of z.  Find the equation of variation when y = 100, x = 2, w = 4, and z = 20. Then find k.

  18. Example • If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = 15.

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