Graphing Exponential Functions Section 4.3 Graphing Exponential Functions

Graphing Exponential Functions with b > 1 Example Graph by hand. Solution List input–output pairs (see table) Input increases by 1 and output multiplies by 2 Plot these points (see next slide) Section 4.3 Slide 2

Graphing Exponential Functions with b > 1 Solution Continued Use graphing calculator to verify Section 4.3 Slide 3

Graphing Exponential Functions with 0< b < 1 Example Graph by hand. Solution List input–output pairs (see table) For example (–1, 8) is a solution x increases by 1, y is multiplied by ½ Section 4.3 Slide 4

Graphing Exponential Functions with 0< b < 1 Solution Continued Section 4.3 Slide 5

Property Illustration Base Multiplier Property; Increase or Decreasing Property Base Multiplier Property Property For an exponential function of the form y = abx, if the value of the independent variable increases by 1, the value of the dependent variable is multiplied by b. For the function , as the value of x increases by 1, the value of y is multiplied by 3 For the function , as the value of x increases by 1, the value of y is multiplied by 3/4 Illustration Section 4.3 Slide 6

Increase or Decrease Property Base Multiplier Property Property Let , 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). Section 4.3 Slide 7

Y-intercept of an Exponential Function Intercepts Property For an exponential function of the form the y-intercept is (0, a). The function , the y-intercept is (0, 5) The function , the y-intercept is (0, 4) Illustration Section 4.3 Slide 8

Intercepts and Graph of an Exponential Function Warning Exponential function of the form , the y- intercept is not (0, b). By writing , we see that the y-intercept is (0, 1). For example, for , the y-intercept is (0, 1). Let 1. Find the y-intercept of f. Example Section 4.3 Slide 9

Intercepts and Graph of an Exponential Function Solution is of the form , 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 Example Solution Section 4.3 Slide 10

Intercepts and Graph of an Exponential Function Solution 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. Example Section 4.3 Slide 11

Plot solutions from the table Intercepts and Graph of an Exponential Function Intercepts Solution Plot solutions from the table Verify on graphing calculator Section 4.3 Slide 12

Finding Values of a Function from Its Graph Reflection Property Example The graph of an exponential function f is shown. Find f(2). Blue arrow shows input of x = 2 leads to an output y = 8 f(2) = 8 Solution Section 4.3 Slide 13

Finding Values of a Function from Its Graph Reflection Property Example 2. Find x when f(x) = 2. Red arrow shows output of y = –2 leads to an input x = 2 x = –2 when f(x) = 2 Solution Section 4.3 Slide 14

Finding Values of a Function from Its Graph Reflection Property Example 3. Find x when f(x) = 0. Graphs of exponential functions get close to zero, but never reaches x-axis No value of x where f(x) = 0 Solution Section 4.3 Slide 15