8 -1 Variation Functions Vocabulary direct variation constant

8 -1 Variation Functions Vocabulary direct variation constant of variation combined variation Holt Algebra 2 inverse variation joint variation

8 -1 Variation Functions Notes 1. The volume V of a pyramid varies jointly as the area of the base B and the height h, and V = 24 ft 3 when B = 12 ft 2 and h = 6 ft. Find B when V = 54 ft 3 and h = 9 ft. 2. The cost person c of chartering a tour bus varies inversely as the number of passengers n. If it costs $22. 50 person to charter a bus for 20 passengers, how much will it cost person to charter a bus for 36 passengers? 3. Determine whether each data set represents a direct variation, an inverse variation, or neither. 3 a. x y Holt Algebra 2 3. 75 15 12 3 5 9 3 b. x y 1 0. 2 40 8 26 5. 2

8 -1 Variation Functions You can use algebra to write variation functions in terms of k. Notice that in direct variation, the ratio of the two quantities is constant. In inverse variation, the product of the two quantities is constant. Holt Algebra 2

8 -1 Variation Functions A direct variation equation is a linear equation in the form y = mx + b, where b = 0 and the constant of variation k is the slope. Because b = 0, the graph of a direct variation always passes through the origin. Holt Algebra 2

8 -1 Variation Functions Example 1: Solving Direct Variation Problems The cost of an item in euros e varies directly as the cost of the item in dollars d, and e = 3. 85 euros when d = $5. 00. Find d when e = 10. 00 euros. Use a proportion. e 1 = e 2 d 1 d 2 3. 85 10. 00 = 5. 00 d 12. 99 ≈ d Holt Algebra 2 Substitute. Solve for d.

8 -1 Variation Functions Example 2: Writing and Graphing Direct Variation Given: y varies directly as x, and y = 27 when x = 6. Write and graph the direct variation function. y = kx y varies directly as x. 27 = k(6) Substitute 27 for y and 6 for x. k = 4. 5 Solve for the constant of variation k. y = 4. 5 x Write the variation function in linear form using the value of k as the slope. Holt Algebra 2

8 -1 Variation Functions Example 2 Continued Graph the direct variation function. The y-intercept is 0, and the slope is 4. 5. Holt Algebra 2

8 -1 Variation Functions Helpful Hint If k is positive in a direct variation, the value of y increases as the value of x increases. A direct variation equation is a linear equation in the form y = mx + b, where b = 0 and the constant of variation k is the slope. Because b = 0, the graph of a direct variation always passes through the origin. Holt Algebra 2

8 -1 Variation Functions A joint variation is a relationship among three variables that can be written in the form y = kxz, where k is the constant of variation. For the equation y = kxz, y varies jointly as x and z. Holt Algebra 2

8 -1 Variation Functions Example 3: Solving Joint Variation Problems The volume V of a cone varies jointly as the area of the base B and the height h, and V = 12 ft 3 when B = 9 ft 3 and h = 4 ft. Find b when V = 24 ft 3 and h = 9 ft. Step 1 Find k. V = k. Bh 12 = k(9 )(4) Substitute. 1 =k Solve for k. 3 Step 2 Use the variation function. 1 V = Bh Use 1 for k. 3 3 1 24 = B(9) Substitute. 3 8 = B The base is 8 ft 2. Holt Algebra 2 Solve for B.

8 -1 Variation Functions A third type of variation describes a situation in which one quantity increases and the other decreases. For example, the table shows that the time needed to drive 600 miles decreases as speed increases. This type of variation is an inverse variation. An inverse variation is a relationship between two variables x and y that can be written in the form y k = , where k ≠ 0. For the equation y = , k y x x varies inversely as x. Holt Algebra 2

8 -1 Variation Functions Example 4: Writing and Graphing Inverse Variation Given: y varies inversely as x, and y = 4 when x = 5. Write and graph the inverse variation function. y= k x 4= k 5 k = 20 y = 20 x Holt Algebra 2 y varies inversely as x. Substitute 4 for y and 5 for x. Solve for k. Write the variation formula.

Variation Functions 8 -1 Example 4 Continued To graph, make a table of values for both positive and negative values of x. Plot the points, and connect them with two smooth curves. Because division by 0 is undefined, the function is undefined when x = 0. x – 2 – 4 – 6 – 8 Holt Algebra 2 y – 10 – 5 – 10 3 5 – 2 x 2 4 6 8 y 10 5 10 3 5 2

8 -1 Variation Functions Example 5 The time t that it takes for a group of volunteers to construct a house varies inversely as the number of volunteers v. If 20 volunteers can build a house in 62. 5 working hours, how many working hours would it take 15 volunteers to build a house? Use t 1 v 1 = k = t 2 v 2. (62. 5)(20) = 1250 = 15 t Substitute and find constant product. Find t. So the number of working hours it would take 15 volunteers to build a house is approximately 83. 3 hours. Holt Algebra 2

Variation Functions 8 -1 Example 7: Identifying Direct and Inverse Variation Determine whether each data set represents a direct variation, an inverse variation, or neither. A. x y C. Holt Algebra 2 6. 5 8 x y 13 104 4 0. 5 3 5 6 14 8 21 B. x y 5 30 8 48 12 72

8 -1 Variation Functions Notes: Part I 1. The volume V of a pyramid varies jointly as the area of the base B and the height h, and V = 24 ft 3 when B = 12 ft 2 and h = 6 ft. Find B when V = 54 ft 3 and h = 9 ft. 18 ft 2 2. The cost person c of chartering a tour bus varies inversely as the number of passengers n. If it costs $22. 50 person to charter a bus for 20 passengers, how much will it cost person to charter a bus for 36 passengers? $12. 50 Holt Algebra 2

8 -1 Variation Functions Notes 3 Determine whether each data set represents a direct variation, an inverse variation, or neither. 3 a. 3 b. x y Holt Algebra 2 3. 75 15 12 3 1 0. 2 40 8 5 9 26 5. 2 In each case xy = 45. The ratio is constant, so this represents an inverse variation. In each case y = 0. 2. x The ratio is constant, so this represents a direct variation.