1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 1 Chapter 4 Exponential Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 2 4.3 Graphing Exponential Functions

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 3 Example: Graphing an Exponential Function with b > 1 Graph f(x) = 2 x by hand.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 4 Solution First, list input-output pairs of the function f in a table. Note that as the value of x increases by 1, the value of y is multiplied by 2 (the base).

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 5 Solution Next, plot the solutions from the table and sketch an increasing curve that contains the plotted points.

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 6 Solution We can set up a window to verify our graph on a graphing calculator.

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 7 Exponential Curve The graph of an exponential function is called an exponential curve.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 8 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.

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 9 Increasing or Decreasing Property Let f(x) = ab x, where a > 0. Then, If b > 1, then the function f is increasing. We say the function grows exponentially. If 0 < b < 1, then the function f is decreasing. We say the function decays exponentially.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 10 y-Intercept of an Exponential Function For an exponential function of the form y = ab x, the y-intercept is (0, a).

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 11 y-Intercept of an Exponential Function Warning For an exponential function of the form y = b x (rather than y = ab x ), the y-intercept is not (0, b). By writing y = b x = 1b x, we see the y-intercept is (0, 1).

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 12 Example: Intercepts and Graph of an Exponential Function Let 1. Find the y-intercept of the graph of f. 2. Find the x-intercept of the graph of f. 3. Graph f by hand.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 13 Solution 1. Since is of the form f(x) = ab x, the y-intercept is (0, a), or (0, 6).

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 14 Solution 2. By the base multiplier property, as the value of x increases by 1, the value of y is multiplied by one half. Values are shown in the table below.

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 15 Solution 2. When we halve a number, it becomes smaller. But no number of halvings will give a result that is zero. So, as x grows large, y will become extremely close to, but never equal, 0. Likewise, the graph of f gets arbitrarily close to, but never reaches, the x-axis. In this case, we call the x-axis a horizontal asymptote. We conclude that the function f has no x-intercepts.

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 16 Solution 3. Plot the points from the table and sketch a decreasing exponential curve that contains the points.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 17 Reflection Property The graphs of f(x) = –ab x and g(x) = ab x are reflections of each other across the x-axis.

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 18 Horizontal Asymptote For all exponential functions, the x-axis is a horizontal asymptote.

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 19 Domain and Range of an Exponential Function The domain of any exponential function f(x) = ab x is the set of real numbers. The range of an exponential function f(x) = ab x is the set of all positive real numbers if a > 0, and the range is the set of all negative real numbers if a < 0.

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 20 Example: Finding Values of a Function from Its Graph The graph of an exponential function f is shown below. 1. Find f(2). 2. Find x when f(x) = 2. 3. Find x when f(x) = 0.

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 21 Solution 1. The blue arrows show that the input x = 2 leads to the output y = 8. We conclude that f(2) = 8. 2. The red arrows show that the output y = 2 originates from the input x = –2. So, x = –2 when f(x) = 2. 3. Recall that the graph of an exponential functions gets close to, but never reaches, the x-axis. So, there is no value of x where f(x) = 0.