366 CHAPTER 5 Exponential and Logarithmic Functions Retain Your Knowledge Problems 31–40 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 31. Find the linear function f whose graph contains the points ( ) 4, 1 and ( ) − 8, 5 . 32. Determine whether the graphs of the linear functions ( ) = − f x x5 1 and ( ) = + g x x 1 5 1 are parallel, perpendicular, or neither. 33. Write the logarithmic expression ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ x y z ln 2 as the sum and / or difference of logarithms. Express powers as factors. 34. Find the domain of ( ) = + + − f x x x x 3 2 8 . 2 35. If ( ) = − − f x x x 2 3 4 and ( ) = + − g x x x 3 1 3 , find ( )( ) −g f x . 36. Find the x -intercept(s) and y -intercept(s) of the graph of ( ) = − + f x x x 2 5 1. 2 37. Solve: + − + = x x x x 1 1 2 38. For the data provided, use a graphing utility to find the line of best fit. What is the correlation coefficient? x −4 −2 0 2 4 6 y 9 5 4 2 −1 −2 39. Use a graphing utility to graph ( ) = − + − f x x x x 3 2 1 4 2 over the interval [ ] −3, 3 . Then, approximate any local maximum values and local minimum values, and determine where f is increasing and where f is decreasing. Round answers to two decimal places. 40. Write ( ) ( ) + + + x x x 10 3 2 3 5 2 3 2 3 1 3 as a single quotient in which only positive exponents appear. OBJECTIVES 1 Build an Exponential Model from Data (p. 367) 2 Build a Logarithmic Model from Data (p. 368) 3 Build a Logistic Model from Data (p. 369) 5.9 Building Exponential, Logarithmic, and Logistic Models from Data • Building Linear Models from Data (Section 3.2, pp. 149–153) • Building Cubic Models from Data (Section 4.2, pp. 210–211) • Building Quadratic Models from Data (Section 3.4, pp. 174–175) PREPARING FOR THIS SECTION Before getting started, review the following: In Section 3.2 we discussed how to find the linear function of best fit ( ) = + y ax b , in Section 3.4 we discussed how to find the quadratic function of best fit ( ) = + + y ax bx c , 2 and in Section 4.2 we discussed how to find the cubic function of best fit ( ) = + + + y ax bx cx d . 3 2 In this section we discuss how to use a graphing utility to find equations of best fit that describe the relation between two variables when the relation is thought to be exponential ( ) = y ab , x logarithmic ( ) = + y a b x ln , or logistic ( ) = + − y c ae 1 . bx As before, we draw a scatter plot of the data to help to determine the appropriate model to use. Figure 60 on the next page shows scatter plots that are typically observed for the three models. Most graphing utilities have REGression options that fit data to a specific type of curve. Once the data have been entered and a scatter plot obtained, the type of curve that you want to fit to the data is selected. Then that REGression option is used to obtain the curve of best fit of the type selected.
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