1. Inverse Variation (11-1) Identify and use inverse variations. Graph inverse variations.

2. Inverse Variation An inverse variation can be represented by the equation y = k / x or xy = k. y varies inversely as x if there is some nonzero constant k such that y = k / x or xy = k, where x, y ≠ 0. In an inverse variation, the product of two values remains constant. Recall that a relationship of the form y = kx is a direct variation. In a direct variation, the quotient of two values remains constant. The constant k is called the constant of variation or the constant of proportionality.

3. Example 1 Determine whether each table or equation represents an inverse or a direct variation. a.. XY 63 84 105 k = 6 3 = 18 Inverse Variation? k = 8 4 = 32 k = 10 5 = 50 NO k = 3 / 6 = 1 / 2 Direct Variation?YES k = 4 / 8 = 1 / 2 k = 5 / 10 = 1 / 2 Direct Variation y = ½ x

4. Example 1 Determine whether each table or equation represents an inverse or a direct variation. b.. XY 112 26 34 k = 1 12 = 12 Inverse Variation? k = 2 6 = 12 k = 3 4 = 12 YES k = 12 / 1 = 12 Direct Variation?NO k = 6 / 2 = 3 k = 4 / 3 Inverse Variation xy = 12

5. Example 1 Determine whether each table or equation represents an inverse or a direct variation. -2xy = 20 Solve for y. -2x y = -10 / x Inverse Variation k = -10 x = 0.5y 0.5 2x = y Direct Variation k = 2 y = 2x c. d.

6. Example 2 You can use xy = k to write an inverse variation equation that relates x and y. – Assume that y varies inversely as x. If y = 5 when x = 3, write an inverse variation equation that relates x and y. xy = k 3 5 = k 15 = k xy = k or y = k / x xy = 15 or y = 15 / x

7. Product Rule for Inverse Variations If (x 1, y 1 ) and (x 2, y 2 ) are solutions of an inverse variation, then the products of x 1 y 1 and x 2 y 2 are equal. x 1 y 1 = x 2 y 2 or

8. Example 3 Assume that y varies inversely as x. If y = 5 when x = 12, find x when y = 15. x 1 y 1 = x 2 y 2 (12)(5) = x 2 (15) 60 = 15x 2 15 4 = x 2 1 1 2 2

9. Example 4 The product rule for inverse variations can be used to write an equation to solve real-world problems. When two people are balanced on a seesaw, their distances from the center of the seesaw are inversely proportional to their weights. How far should a 105-pound person sit from the center of the seesaw to balance a 63-pound person sitting 3.5 feet from the center? y = distance (ft) x = weight (lbs) x 1 y 1 = x 2 y 2 (63)(3.5) = (105)y 2 220.5 = 105y 2 105 2.1 ft = y 2

10. Graph Inverse Variations The graph of an inverse variation is not a straight line like the graph of a direct variation.

11. Example 5 Graph an inverse variation equation in which y = 1 when x = 4. xy = k 4 1 = k 4 = k xy = 4 or y = 4 / x XY -8 -4 -2 0 1 2 4 8 -½ -2 -4 undef 4 2 1 ½

12. Summary Direct Variation y = kx y varies directly as x The ratio y / x is a constant x y y = kx k > 0 x y y = kx k < 0

13. Summary Inverse Variation y = k / x y varies inversely as x The product xy is a constant x y y = k / x k > 0 x y y = k / x k < 0

14. Check Your Progress Choose the best answer for the following. A.Determine whether the table represents an inverse or a direct variation. A.direct variation B.inverse variation XY 120 210 45 xy = 20

15. Check Your Progress Choose the best answer for the following. B.Determine whether the table represents an inverse or a direct variation. A.direct variation B.inverse variation XY 29 418 627 y / x = 9 / 2

16. Check Your Progress Choose the best answer for the following. C.Determine whether 2x = 4y represents an inverse or a direct variation. A.direct variation B.inverse variation 2x = 4y 4 ½ x = y

17. Check Your Progress Choose the best answer for the following. D.Determine whether x = 6 / y represents an inverse or a direct variation. A.direct variation B.inverse variation xy = 6 x y = 6 / x

18. Check Your Progress Choose the best answer for the following. – Assume that y varies inversely as x. If y = -3 when x = 8, determine a correct inverse variation equation that relates x and y. A.-3y = 8x B.xy = 24 C.y = -24 / x D.y = x / -24 xy = k 8 -3 = k -24 = k xy = k or y = k / x

19. Check Your Progress Choose the best answer for the following. – If y varies inversely as x and y = 6 when x = 40, find x when y = 30. A.5 B.20 C.8 D.6 x 1 y 1 = x 2 y 2 (40)(6) = x 2 (30) 240 = 30x 2 30 1 1 2 2

20. Check Your Progress Choose the best answer for the problem. – When two objects are balanced on a lever, their distances from the fulcrum are inversely proportional to their weights. How far should a 2-kilogram weight be from the fulcrum if a 6-kilogram weight is 3.2 meters from the fulcrum? A.2 m B.3 m C.4 m D.9.6 m w = weight (kg) d = distance (m) w 1 d 1 = w 2 d 2 (6)(3.2) = (2)d 2 19.2 = 2d 2 2

21. Check Your Progress Choose the best answer for the following. – Graph an inverse variation equation in which y = 8 when x = 3. xy = k 3 8 = k 24 = k xy = 24 or y = 24 / x XY -12 -6 -2 0 2 6 12 -2 -4 -12 undef 12 4 2 A. B. C. D.