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Graphing Equations and Inequalities

Chapter 10. Graphing Equations and Inequalities. Direct and Inverse Variation. 10.8. 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

walter-hebert
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Graphing Equations and Inequalities

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  1. Chapter 10 Graphing Equations and Inequalities

  2. Direct and Inverse Variation 10.8

  3. 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 orthe constant of proportionality.

  4. Direct Variation 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 = So the direct variation equation is

  5. Example 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

  6. Direct Variation 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. Let x = 0. Then y = k∙ 0 or y = 0.

  7. Example y x The line is the graph of a direct variation equation. Find the constant of variation and the direct variation equation. (4, 1) To find k, use the slope formula and find slope. (0 0) and the variation equation is

  8. Inverse Variation 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. x

  9. Example 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 So the inverse variation equation is

  10. Powers of x y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that 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 = kxn

  11. Example 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

  12. continued Translate the problem into an equation. Substitute the given values for the elevation and distance to the horizon for e and d. Simplify. Solve for k, the constant of proportionality. continued

  13. continued So the equation is . Replace e with 64. Simplify. A person 64 feet above the water can see about 9.87 miles.

  14. Example 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

  15. continued Translate the problem into an equation. Substitute the given values for w and h. Solve for k, the constant of proportionality. So the equation is . Let h = 10. A 10-foot column can hold 1.28 tons.

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