Chapter 4 More Nonlinear Functions and Equations

1. Chapter 4 More Nonlinear Functions and Equations

2. Radical Equations and Power Functions 4.8 Learn properties of rational exponents Learn radical notation Use radical functions and solve radical equations Understand properties and graphs of power functions Use power functions to model data Solve equations involving rational exponents Use power regression to model data (optional)

3. Properties of Rational Exponents Let m and n be positive integers with m/n in lowestterms and n ≥ 2. Let r and p berational numbers. Assume that b is a nonzero real number and that each expression isa real number. Property and Example

4. Properties of Rational Exponents z

5. Properties of Rational Exponents z

6. Properties of Rational Exponents z

7. Simplify each expression by hand. (a) 163/4 (b) 27-2/3•271/3 (c) (–125)–4/3 Solutions Example: Applying properties of exponents

8. Use positive rational exponents to write each expression. Solutions Example: Writing radicals with rational exponents

9. Equations Involving Radicals When solving equations that contain square roots, it is common to square each side of an equation.

10. Solve Check your answers. Solution Example: Solving an equation containing a square root

11. Check: Substituting these values in the original equation shows that the value of –5 is an extraneous solution because it does not satisfy the given equation. Therefore, the only solution is 3. Example: Solving an equation containing a square root

12. Some equations require squaring twice. Solve Solution Example: Squaring twice

13. Both solutions check, so the solution set is {–1, 3}. Example: Squaring twice

14. Power Functions and Models Power functions typically have rational exponents. A special type of power function is a root function. Examples of power functions include: f1(x) = x2f2(x) = x3/4 , f3(x) = x0.4

15. Power Functions A function f given by f(x) = xb, where b is a constant, is a power function. If b = 1/n for some integer n ≥ 2, then f is a root function given by f(x) = x1/n, or equivalently,

16. Graph f(x) = xb, where b = 1, and 1.7, for x ≥ 0. Discuss the effect that b has onthe graph of f for x ≥ 1. Example: Graphing power functions • Solution • Larger values of b cause the graph of f to increase faster.

17. Heavier birds have larger wings with more surface area than do lighter birds. For somespecies of birds, this relationship can be modeled by S(w) = 0.2w2/3, where w is theweight of the bird in kilograms, with0.1 ≤ w ≤ 5, and S is the surface area of the wingsin square meters. (a)Approximate Sand interpret the result. (b)What weight corresponds to a surface area of square meter? Example: Modeling wing size of a bird

18. Solution (a) S(0.5) = 0.2(0.5)2/3 ≈ 0.126. The wings of a bird that weighs 0.5 kilogram have asurface area of about 0.126 square meter. (b) Solve Example: Modeling wing size of a bird

19. Since w must be positive, the wings of a 1.4-kilogram bird have a surface area of about0.25square meter. Example: Modeling wing size of a bird

20. Equations Involving Rational Exponents Equations sometimes have rational exponents. The next example demonstrates a basictechnique that can be used to solve some of these types of equations.

21. Solve 2x5/2 – 7 = 23. Round to the nearest hundredth, and give graphical support. Solution Symbolic Solution Graphical Solution 2x5/2 = 30 x5/2 = 15 (x5/2)2 = 152 x5 = 225 x = 2251/5 x ≈ 2.95 Example: Solving an equation with rational exponents

22. Solve Solution Example: Solve an equation having negative exponents

23. Power Regression Rather than visually fit a curve to data, we can use least-squaresregression to fit the data. Least-squares regression was introduced in Section 2.1. In thenext example, we apply this technique to data from biology.

24. The table lists the weight W and the wingspan L for birds of a particular species. (a)Use power regression to model the data with L = aWb.Graph the data and the equation. (b)Approximate the wingspan for a bird weighing 3.2 kilograms. Example: Modeling the length of a bird’s wing

25. Solution Let x be the weight W and y be the length L. Enter the data and select power regression. Example: Modeling the length of a bird’s wing

26. The results shown yield: y = 0.9674x0.3326 or L = 0.9674W0.3326 Example: Modeling the length of a bird’s wing

27. (b) If a bird weighs 3.2kilograms, this model predicts the wingspan to be L = 0.9674(3.2)0.3326 ≈ 1.42 meters Example: Modeling the length of a bird’s wing