1. Direct and Inverse Variation 1.You will write and graph direct variation equations. 2.You will use inverse variation and joint variation models.

3. Write and graph a direct variation equation Write and graph a direct variation equation that has (– 4, 8) as a solution. SOLUTION Use the given values of x and y to find the constant of variation. y = ax 8 = a(– 4) –2= a Substituting – 2 for a in y = ax gives the direct variation equation y = – 2x. Its graph is shown.

4. Try this. Write and graph a direct variation equation that has the given ordered pair as a solution. (3, – 9) Use the given values of x and y to find the constant of variation. y = ax –9 = a(3) –3= a Substituting – 3 for a in y = ax gives the direct variation equation y = – 3x. Its graph is shown.

5. Use the given values of x and y to find the constant of variation. Try this. Write and graph a direct variation equation that has the given ordered pair as a solution. 2. (– 7, 4) y = ax 4 = a(– 7) – 4 7 = a

6. Write and apply a model for direct variation Meteorology Hailstones form when strong updrafts support ice particles high in clouds, where water droplets freeze onto the particles. The diagram shows a hailstone at two different times during its formation. a. Write an equation that gives the hailstone’s diameter d (in inches) after t minutes if you assume the diameter varies directly with the time the hailstone takes to form. b. Using your equation from part (a), predict the diameter of the hailstone after 20 minutes.

7. Write and apply a model for direct variation a. Use the given values of t and d to find the constant of variation. d = atd = at 0.75 = a(12) 0.0625 = a An equation that relates t and d is d = 0.0625t. b. After t = 20 minutes, the predicted diameter of the hailstone is d = 0.0625(20) = 1.25 inches.

8. Use ratios to identify direct variation Tooth length, t (cm) 1.82.42.93.64.75.8 Body length, b (cm) 215290350430565695 Find the ratio of the body length b to the tooth length t for each shark. 215 1.8 119 290 2.4 121 430 3.6 119 565 4.7 120 695 5.8 120 350 2.9 121 = Because the ratios are approximately equal, the data show direct variation. An equation relating tooth length and body length is = 120, or b = 120t.

10. Classify direct and inverse variation Tell whether x and y show direct variation, inverse variation, or neither. Given EquationRewritten Equation Type of Variation a. xy = 7 y = 7 x Inverse b. y = x + 3 Neither y 4 c. = x y = 4x Direct

11. Write an inverse variation equation The variables x and y vary inversely, and y = 7 when x = 4. Write an equation that relates x and y. Then find y when x = –2. y = y = a x 7 = 7 = a 4 28 = a The inverse variation equation is y = 28 x When x = –2, y = 28 –2 = –14. ANSWER

12. Write an inverse variation model The number of songs that can be stored on an MP3 player varies inversely with the average size of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB). MP3 Players Write a model that gives the number n of songs that will fit on the MP3 player as a function of the average song size s ( in megabytes ). Make a table showing the number of songs that will fit on the MP3 player if the average size of a song is 2MB, 2.5MB, 3MB, and 5MB as shown below. What happens to the number of songs as the average song size increases?

13. Write an inverse variation model Write an inverse variation model. a n =n = s a 2500 = 4 10,000 = a A model is n = s 10,000 ANSWER Make a table of values. From the table, you can see that the number of songs that will fit on the MP3 player decreases as the average song size increases.

14. Try these. Tell whether x and y show direct variation, inverse variation, or neither. 1. 3x = y 2. xy = 0.75 3. y = x –5 Given EquationRewritten Equation Type of Variation y = 3x Direct y = x 0.75 Inverse Neither

15. Try this. 4. x = 4, y = 3 The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 2. y = y = a x 4 3 = 3 = a 12 = a 5. x = 8, y = –1 a y = x –1 = a 8 – 8 = a

16. Check data for inverse variation The table compares the area A ( in square millimeters) of a computer chip with the number c of chips that can be obtained from a silicon wafer. Computer Chips Write a model that gives c as a function of A. Predict the number of chips per wafer when the area of a chip is 81 square millimeters.

17. Check data for inverse variation Calculate the product A c for each data pair in the table. 58(448) = 25,984 62(424) = 26,288 66(392) = 25,872 70(376) = 26,320 Each product is approximately equal to 26,000. So, the data show inverse variation. A model relating A and c is: A c = 26,000, or c = A 26,000 Make a prediction. The number of chips per wafer for a chip with an area of 81 square millimeters is 81 26,000 321 c =

18. Write a joint variation equation The variable z varies jointly with x and y. Also, z = –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find z when x = 2 and y = 6. Write a general joint variation equation. z = axy –75 = a(3)(–5) Use the given values of z, x, and y to find the constant of variation a. –75 = –15a 5 = a Rewrite the joint variation equation with the value of a z = 5xy Calculate z when x = 2 and y = 6 using substitution. z = 5xy = 5(2)(6) = 60

19. Try this. The variable z varies jointly with x and y. Use the given values to write an equation relating x, y, and z. Then find z when x = –2 and y = 5. x = 1, y = 2, z = 7 Write a general joint variation equation. z = axy 7 = a(1)(2) Use the given values of z, x, and y to find the constant of variation a. 7 = 2a Rewrite the joint variation equation with the value of a = a 7 2 z = xy 7 2 Calculate z when x = – 2 and y = 5 using substitution. z = xy = (– 2)(5) = – 35 7 2 7 2 z = xy 7 2 ; – 35

20. Compare different types of variation Write an equation for the given relationship. Relationship Equation a. y varies inversely with x. b. z varies jointly with x, y, and r. z = axyr y = a x c. y varies inversely with the square of x. y = a x2x2 d. z varies directly with y and inversely with x. z = ay x e. x varies jointly with t and r and inversely with s. x = atr s

21. Try this. Write an equation for the given relationship. x varies inversely with y and directly with w. p varies jointly with q and r and inversely with s. x = a y w p = aqr s

22. ASSIGNMENT P. 109: 10, 16, 20, 23, 28, 31, 32, 38 P. 555: 19, 21, 23, 28, 33, 37