1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 10 Graphing Equations and Inequalities

2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.8 Direct and Inverse Variation

3. Martin-Gay, Developmental Mathematics, 2e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Direct Variation y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that y = kx The number k is called the constant of variation or the constant of proportionality.

4. Martin-Gay, Developmental Mathematics, 2e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Suppose that y varies directly as x. If y = 5 when x = 30, find the constant of variation and the direct variation equation. y = kx 5 = k 30 k = Direct Variation So the direct variation equation is

5. Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Suppose that y varies directly as x, and y = 48 when x = 6. Find y when x = 15. y = kx 48 = k 6 8 = k So the equation is y = 8x. y = 8 ∙ 15 y = 120 Example

6. Martin-Gay, Developmental Mathematics, 2e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Direct Variation: y = kx There is a direct variation relationship between x and y. The graph is a line. The line will always go through the origin (0, 0). Why? The slope of the graph of y = kx is k, the constant of variation. Why? Remember that the slope of an equation of the form y = mx + b is m, the coefficient of x. The equation y = kx describes a function. Each x has a unique y and its graph passes the vertical line test. Direct Variation Let x = 0. Then y = k ∙ 0 or y = 0.

7. Martin-Gay, Developmental Mathematics, 2e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The line is the graph of a direct variation equation. Find the constant of variation and the direct variation equation. Example x y (0 0) (4, 1) To find k, use the slope formula and find slope. and the variation equation is

8. Martin-Gay, Developmental Mathematics, 2e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that y = k The number k is called the constant of variation or the constant of proportionality. Inverse Variation x

9. Martin-Gay, Developmental Mathematics, 2e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Suppose that y varies inversely as x. If y = 63 when x = 3, find the constant of variation k and the inverse variation equation. k = 63·3 k = 189 Example So the inverse variation equation is

10. Martin-Gay, Developmental Mathematics, 2e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Direct and Inverse Variation as nth Powers of x y varies directly as a power of x if there is a nonzero constant k and a natural number n such that y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that Powers of x y = kx n

11. Martin-Gay, Developmental Mathematics, 2e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places. continued Example

12. Martin-Gay, Developmental Mathematics, 2e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Substitute the given values for the elevation and distance to the horizon for e and d. continued Simplify. Solve for k, the constant of proportionality. Translate the problem into an equation. continued

13. Martin-Gay, Developmental Mathematics, 2e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. So the equation is Replace e with 64. Simplify. A person 64 feet above the water can see about 9.87 miles.. continued

14. Martin-Gay, Developmental Mathematics, 2e 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The maximum weight that a circular column can hold is inversely proportional to the square of its height. If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold. continued Example

15. Martin-Gay, Developmental Mathematics, 2e 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. So the equation is continued Substitute the given values for w and h. Solve for k, the constant of proportionality. Translate the problem into an equation. A 10-foot column can hold 1.28 tons. Let h = 10..