Direct and Inverse Variations Lesson 9-1 Direct and Inverse Variations 1

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.

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

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

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

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

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

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. 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

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

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

continued So the equation is A 10-foot column can hold 1.28 tons. 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.

Direct, Inverse, or Neither  

Direct or Inverse? Is the relationship in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations? A. x 0.5 2 6 y 1.5 18

Direct or Inverse? Is the relationship in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations? B. x 0.2 0.6 1.2 y 12 4 2

Direct or Inverse? Is the relationship in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations? C. x 1 2 3 y 0.5

Write the Inverse Function Heart rates and life spans of most mammals are inversely related. Write the inverse function for the table. Use your function to estimate the average life span of a cat with a heart rate of 126 beats/min. A squirrel’s heart rate is 190 beats/min. Estimate it’s life span. An elephant’s life span is about 70 years. Estimate it’s average heart rate. Mammal Heart Rate (beats/min) Life Span (min) Mouse 634 1,576,300 Rabbit 158 6,307,200 Lion 76 13,140,000 Horse 63 15,768,000

Combined Variations It is possible for three or more variables to be related. When one quantity varies with respect to two or more other quantities, the result is a combined variation. Combined Variation Equation Form Z varies jointly with x and y. Z varies jointly with x and y and inversely with w. Z varies directly with x and inversely with the product wy.

Writing Combined Variation  

Finding a Formula  

Homework: L9-1 (p 499) #8-26e 36-56e