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Objectives Direct and Inverse Variation Direct Variation – Solve direct variation problems. – Solve inverse variation problems. – Solve combined variation problems. – Solve problems involving joint variation. Direct variation COULD involve an nth power of x (no longer linear) • y varies directly as x (y is directly proportional to x) if y = kx. • k is the constant of variation (constant of proportionality) • Still involves a constant of proportionality n • This is the graph of a linear function with slope = m, crossing through the origin. • y is directly proportional to the nth power of x. y = kx • Example: You are paid $8/hr. Thus, pay is directly related to hours worked: pay=8(hours worked) Solving Variation Problems 1. Write an equation that describes the given English statement. 2. Substitute the given pair of values into the equation in step 1 and solve for k, the constant of variation. 3. Substitute the value of k into the equation in step 1. 4. Use the equation in step 3 to answer the problem’s question. Example • An object’s weight on the moon, M, varies directly as its weight on Earth, E. Neil Armstrong, the first person to step on the moon on July 20, 1969, weighed 360 pounds on Earth (with all his equipment on) and 60 pounds on the moon. What is the moon weight of a person who weighs 186 pounds on Earth? 1 Inverse Variation Example • The water temperature of the Pacific Ocean varies inversely as the water’s depth. At a depth of 1000 meters, the water temperature is 4.4° Celsius. What is the water temperature at a depth of 5000 meters? k y= x • y varies inversely as x or y is inversely proportional to x. • k is the constant of variation. • As x gets bigger, y gets smaller • As x gets smaller, y gets bigger Combined Variation Problem • y is impacted by TWO variables in TWO different ways. One variable causes y to get bigger, while the other variable causes it to become smaller. y= k⋅x z Example • y varies jointly as m and the square of n and inversely as p. y = 15 when m = 2, n = 1, and p = 6. Find y when m = 3, n = 4, and p = 10. y varies directly as x and inversely as z • As x gets bigger, y gets bigger, but as z gets bigger, y gets smaller. • k must take into account both influences 2