Graphing Exponential Functions
Graphing Exponential Functions with b > 1 Graph f x = ( ) 2 x Graphing Exponential Functions by hand. List input output pairs (see table) Input increases by 1 and output multiplies by 2 Plot these points (see next slide) Slide 2
Graphing Exponential Functions with b > 1 Continued Use graphing calculator to verify Graphing Exponential Functions Slide 3
Graphing Exponential Functions with 0< b < 1 Graphing Exponential Functions x 1 Graph g( x) = 4 by hand. 2 List input output pairs (see table) For example ( 1, 8) is a solution x increases by 1, y is multiplied by ½ Slide 4
Graphing Exponential Functions with 0< b < 1 Continued Graphing Exponential Functions Slide 5
Base Multiplier Property; Increase or Decreasing Property Property Base Multiplier Property For an exponential function of the form y = ab x, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b. Illustration For the function f ( x ) = 23 ( ) x, as the value of x increases by 1, the value of y is multiplied by 3 x 3 For the function f ( x) = 5, as the value of x 4 increases by 1, the value of y is multiplied by 3/4 Slide 6
Increase or Decrease Property Property Base Multiplier Property Let f x= ab, where a > 0. Then If b > 1, then the function f is increasing. We say that the function grows exponentially (left). If 0 < b < 1, then the function f is decreasing. We say that the function decays exponentially (right). ( ) ( ) x Slide 7
Y-intercept of an Exponential Function Property Intercepts For an exponential function of the form y= ab ( ) x the y-intercept is (0, a). Illustration The function f ( x ) = 58 ( ) x, the y-intercept is (0, 5) x 1 The function f ( x) = 4, the y-intercept is (0, 4) 7 Slide 8
Intercepts and Graph of an Exponential Function Warning Intercepts Exponential function of the form y = b, the y- intercept is not (0, b). By writing ( ) x x y = b = 1b, we see that the y-intercept is (0, 1). For example, for y = 2 x, the y-intercept is (0, 1). x 1 Let f ( x) = 6 2 1. Find the y-intercept of f. ( ) x Slide 9
Intercepts and Graph of an Exponential Function Intercepts x 1 f ( x) = 6 is of the form f ( x) = ab ( ) x, 2 We know that the y-intercept is (0, a), or (0, 6). 2. Find the x-intercept of f. By base multiplier property, x increases by 1, y value multiplies by ½ Slide 10
Intercepts and Graph of an Exponential Function Continued No number of halvings will result in zero As x grows large, y gets closer to the x-axis Called horizontal asymptote 3. Graph f by hand. Intercepts Slide 11
Intercepts and Graph of an Exponential Function Intercepts Plot solutions from the table Verify on graphing calculator Slide 12
Finding Values of a Function from Its Graph The graph of an exponential function f is shown. 1. Find f(2). Reflection Property Blue arrow shows input of x = 2 leads to an output y = 8 f(2) = 8 Slide 13
Finding Values of a Function from Its Graph 2. Find x when f(x) = 2. Reflection Property Red arrow shows output of y = 2 leads to an input x = 2 x = 2 when f(x) = 2 Slide 14
Finding Values of a Function from Its Graph 3. Find x when f(x) = 0. Reflection Property Graphs of exponential functions get close to zero, but never reaches x-axis No value of x where f(x) = 0 Slide 15