SECTION 5.3 Exponential Functions 297 In Words For 1-unit changes in the input x of an exponential function ( ) = ⋅ f x C a ,x the ratio of consecutive outputs is the constant a. Identifying Linear or Exponential Functions Determine whether the given function is linear, exponential, or neither. For those that are linear, find a linear function that models the data. For those that are exponential, find an exponential function that models the data. EXAMPLE 2 x y f x( ) = Average Rate of Change Ratio of Consecutive Outputs −1 32 ( ) Δ Δ = − − − = − y x 16 32 0 1 16 ( ) ( ) − = = f f 0 1 16 32 1 2 0 16 −8 ( ) ( ) = = f f 1 0 8 16 1 2 1 8 −4 ( ) ( ) = = f f 2 1 4 8 1 2 2 4 −2 ( ) ( ) = = f f 3 2 2 4 1 2 3 2 (b) (a) x y −1 5 0 2 1 −1 2 −4 3 −7 (b) x y −1 32 0 16 1 8 2 4 3 2 (c) x y −1 2 0 4 1 7 2 11 3 16 Solution For each function, compute the average rate of change of y with respect to x and the ratio of consecutive outputs. If the average rate of change is constant, then the function is linear. If the ratio of consecutive outputs is constant, then the function is exponential. (a) See Table 2(a). The average rate of change for every 1-unit increase in x is −3. Therefore, the function is a linear function. In a linear function the average rate of change is the slope m, so = − m 3. The y -intercept b is the value of the function at = x 0, so = b 2. The linear function that models the data is ( ) = + = − + f x mx b x3 2. x y f x( ) = Average Rate of Change Ratio of Consecutive Outputs −1 5 ( ) Δ Δ = − − − = − y x 2 5 0 1 3 ( ) ( ) − = f f 0 1 2 5 0 2 − − − = − 1 2 1 0 3 ( ) ( ) = − = − f f 1 0 1 2 1 2 1 −1 ( ) − − − − = − 4 1 2 1 3 ( ) ( ) = − − = f f 2 1 4 1 4 2 −4 7 4 3 2 3 ( ) − − − − = − f f 3 2 7 4 7 4 ( ) ( ) = − − = 3 −7 Table 2(a) (continued) Proof ( ) ( ) + = = = = + + − f x f x Ca Ca a a a 1 x x x x 1 1 1 ■
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