Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 1 Chapter 7 Functions and Graphs

2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Formulas, Applications, and Variation Formulas Direct Variation Inverse Variation Joint and Combined Variation 7.5

3 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Formulas Formulas occur frequently as mathematical models. Many formulas contain rational expressions, and to solve such formulas for a specified letter, we proceed as when solving rational equations.

4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example In a hydraulic system, a fluid is confined to two connecting chambers. The pressure in each chamber is the same and is given by finding the force exerted (F) divided by the surface area (A). Therefore, we know Solve for A 2.

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 5 To Solve a Rational Equation for a Specified Variable 1.Multiply both sides by the LCD to clear fractions, if necessary. 2.Multiply to remove parentheses, if necessary. 3.Get all terms with the specified variable alone on one side. 4.Factor out the specified variable if it is in more than one term. 5.Multiply or divide on both sides to isolate the specified variable.

6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Variation To extend our study of formulas and functions, we now examine three real- world situations: direct variation, inverse variation, and combined variation.

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 7 Direct Variation When a situation is modeled by a linear function of the form f (x) = kx, or y = kx, where k is a nonzero constant, we say that there is direct variation, that y varies directly as x, or that y is proportional to x. The number k is called the variation constant, or constant of proportionality.

8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Direct Variation A mass transit driver earns $17 per hour. In 1 hr, $17 is earned. In 2 hr, $34 is earned. In 3 hr, $51 is earned, and so on. This gives rise to a set of ordered pairs: (1, 17), (2, 34), (3, 51), (4, 68), and so on. Note that the ratio of earnings E to time t is 17/1 in every case. If a situation gives rise to pairs of numbers in which the ratio is constant, we say that there is direct variation. Here earnings vary directly as the time: We have E/t = 17, so E = 17t, or using function notation, E(t) = 17t.

9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find the variation constant and an equation of variation if y varies directly as x, and y = 15 when x = 3.

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Inverse Variation When a situation gives rise to a rational function of the form f (x) = k/x, or y = k/x, where k is a nonzero constant, we say that there is inverse variation, that is y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality.

11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example The time, t, required to empty a tank varies inversely as the rate, r, of pumping. If a pump can empty a tank in 90 minutes at the rate of 1080 kL/min, how long will it take the pump to empty the same tank at the rate of 1500 kL/min?

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Joint Variation When a variable varies directly with more than one other variable, we say that there is joint variation. y varies jointly as x and z if, for some nonzero constant k, y = kxz.

13 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Joint and Combined Variation For example, in the formula for the volume of a right circular cylinder, V = πr 2 h, we say that V varies jointly as h and the square of r.

14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find an equation of variation if a varies jointly as b and c, and a = 48 when b = 4 and c = 2.

15 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Joint variation is one form of combined variation. In general, when a variable varies directly and/or inversely, at the same time, with more than one other variable, there is combined variation.

16 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find an equation of variation if y varies jointly as x and z and inversely as the product of w and p, and y = 60 when x = 24, z = 5, w = 2, and p = 3.