298 CHAPTER 5 Exponential and Logarithmic Functions (b) See Table 2(b) on the previous page. For this function, the average rate of change is not constant. So the function is not a linear function. The ratio of consecutive outputs for a 1-unit increase in the inputs is a constant, 1 2 . Because the ratio of consecutive outputs is constant, the function is an exponential function with growth factor = a 1 2 . The initial value C of the exponential function is = C 16, the value of the function at 0. Therefore, the exponential function that models the data is ( ) ( ) = = ⋅ g x Ca 16 1 2 . x x Note that the ratio of consecutive outputs is not constant. (c) See Table 2(c). For this function, neither the average rate of change nor the ratio of two consecutive outputs is constant. Because the average rate of change is not constant, the function is not a linear function. Because the ratio of consecutive outputs is not a constant, the function is not an exponential function. x y f x( ) = Average Rate of Change Ratio of Consecutive Outputs −1 2 ( ) Δ Δ = − − − = y x 4 2 0 1 2 ( ) ( ) − = f f 0 1 2 0 4 3 ( ) ( ) = f f 1 0 7 4 1 7 4 ( ) ( ) = f f 2 1 11 7 2 11 5 ( ) ( ) = f f 3 2 16 11 3 16 Table 2(c) Now Work PROBLEM 31 2 Graph Exponential Functions If we know how to graph an exponential function of the form ( ) = f x a ,x then we can use transformations (shifting, stretching, and so on) to obtain the graph of any exponential function. First, let’s graph the exponential function ( ) = f x 2 .x Graphing an Exponential Function Graph the exponential function: ( ) = f x 2x EXAMPLE 3 Solution The domain of ( ) = f x 2x is the set of all real numbers. Begin by locating some points on the graph of ( ) = f x 2 ,x as listed in Table 3 using a TI-84 Plus CE. Because > 2 0 x for all x, the graph has no x-intercepts and lies above the x-axis for all x. The y-intercept is 1. Table 3 suggests that as →−∞ x , the value of f approaches 0. Therefore, the x-axis ( ) = y 0 is a horizontal asymptote of the graph of f as →−∞ x . This provides the end behavior as x becomes unbounded in the negative direction. To determine the end behavior for x unbounded in the positive direction, look again at Table 3. As ( ) →∞ = x f x , 2x grows very quickly, causing the graph of ( ) = f x 2x to rise very rapidly. Table 3
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