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Holt Algebra 2 8-1 Variation Functions direct variationinverse variation constant of variationjoint variation combined variation Vocabulary

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Holt Algebra 2 8-1 Variation Functions 3. Determine whether each data set represents a direct variation, an inverse variation, or neither. 3a. x3.75155 y1239 Notes 3b. x14026 y0.285.2 2. The cost per person c of chartering a tour bus varies inversely as the number of passengers n. If it costs $22.50 per person to charter a bus for 20 passengers, how much will it cost per person to charter a bus for 36 passengers? 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.

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Holt Algebra 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.

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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.

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Holt Algebra 2 8-1 Variation Functions Substitute. Use a proportion. e1e1 = d1d1 d2d2 e2e2 3.85 = 5.00d 10.00 Solve for d. 12.99 d 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.

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Holt Algebra 2 8-1 Variation Functions Given: y varies directly as x, and y = 27 when x = 6. Write and graph the direct variation function. Example 2: Writing and Graphing Direct Variation y varies directly as x. Substitute 27 for y and 6 for x. Solve for the constant of variation k. Write the variation function in linear form using the value of k as the slope. y = kx 27 = k(6) k = 4.5 y = 4.5x

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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.

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Holt Algebra 2 8-1 Variation Functions If k is positive in a direct variation, the value of y increases as the value of x increases. Helpful Hint 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.

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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.

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Holt Algebra 2 8-1 Variation Functions 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. Example 3: Solving Joint Variation Problems Step 1 Find k. The base is 8 ft 2. Substitute. Solve for k. V = kBh 12 = k(9)(4) = k 1 3 Substitute. Solve for B. Step 2 Use the variation function. 24 = B(9) 8 = B V = Bh 1 3 1 3 Use for k. 1 3

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Holt Algebra 2 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 =, where k 0. For the equation y =, y varies inversely as x. k x k x

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

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Holt Algebra 2 8-1 Variation Functions 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. Example 4 Continued xy –2–10 –4–5 –6– –8– xy 210 45 6 8 3 5 2 5 2 3

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Holt Algebra 2 8-1 Variation Functions Use t 1 v 1 = k = t 2 v 2. 1250 = 15t (62.5)(20) = 1250 Find t. Substitute and find constant product. So the number of working hours it would take 15 volunteers to build a house is approximately 83.3 hours. 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?

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Holt Algebra 2 8-1 Variation Functions Example 7: Identifying Direct and Inverse Variation Determine whether each data set represents a direct variation, an inverse variation, or neither. A. x6.513104 y840.5 B. x5812 y304872 x368 y51421 C.

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Holt Algebra 2 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 per person c of chartering a tour bus varies inversely as the number of passengers n. If it costs $22.50 per person to charter a bus for 20 passengers, how much will it cost per person to charter a bus for 36 passengers? $12.50

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Holt Algebra 2 8-1 Variation Functions Determine whether each data set represents a direct variation, an inverse variation, or neither. 3a. x3.75155 y1239 In each case xy = 45. The ratio is constant, so this represents an inverse variation. Notes 3 3b. x14026 y0.285.2 In each case = 0.2. The ratio is constant, so this represents a direct variation. y x

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