126 CHAPTER 2 Functions and Their Graphs Problems 102–110 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. 102. Find the slope and y -intercept of the line x y 3 5 30 − = 103. Angie runs 7 mph for the first half of a marathon, 13.1 miles, but twists her knee and must walk 2 mph for the second half. What was her average speed? Round to 2 decimal places. 104. The amount of water used when taking a shower varies directly with the number of minutes the shower is run. If a 4-minute shower uses 7 gallons of water, how much water is used in a 9-minute shower? 105. Find the intercepts and test for symmetry: y x 4 2 = + 106. Find the domain of h x x x x 2 5 14 . 2 ( ) = + − − 107. Projectile Motion A ball is thrown upward from the top of a building. Its height h , in feet, after t seconds is given by the equation h t t 16 96 200. 2 = − + + How long will it take for the ball to be 88 ft above the ground? 108. Simplify x y z 16 . 5 6 3 109. Find the difference quotient of f x x x 3 2 1. 2 ( ) = + − 110. Factor z 216. 3 + Retain Your Knowledge Need to Review? The distance formula is covered in Section 1.2, pp. 13 – 16 . Finding the Distance from the Origin to a Point on a Graph Let P x y , ( ) = be a point on the graph of y x 1. 2 = − (a) Graph f. Express the distance d from P to the origin O as a function of x. (b) What is d if x 0? = (c) What is d if x 1? = (d) What is d if x 2 2 ? = (e) Use a graphing utility to graph the function d d x x , 0. ( ) = ≥ Rounding to two decimal places, find the value(s) of x at which d has a local minimum. [This gives the point(s) on the graph of y x 1 2 = − closest to the origin.] EXAMPLE 1 2.6 Mathematical Models: Building Functions OBJECTIVE 1 Build and Analyze Functions (p. 126) 1 Build and Analyze Functions Real-world problems often result in mathematical models that involve functions. These functions need to be constructed or built based on the information given. In building functions, we must be able to translate the verbal description into the language of mathematics. This is done by assigning symbols to represent the independent and dependent variables and then by finding the function or rule that relates these variables. Solution (a) Figure 67 illustrates the graph of y x 1. 2 = − The distance d from P to O is d x y x y 0 0 2 2 2 2 ( ) ( ) = − + − = + Since P is a point on the graph of y x 1, 2 = − substitute x 1 2 − for y. Then d x x x x x 1 1 2 2 2 4 2 ( ) ( ) = + − = − + The distance d is expressed as a function of x. (b) If x 0, = the distance d is d 0 0 0 1 1 1 4 2 ( ) = − + = =
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