1. Copyright © 2011 Pearson Education, Inc. Slide 5.2-1 5.2 Exponential Functions Additional Properties of Exponents For any real number a > 0, a  0, the following statements are true: (a) a x is a unique real number for each real number x. (b) a b = a c if and only if b = c. (c) If a > 1 and m < n, then a m < a n. (d) If 0 a n.

2. Copyright © 2011 Pearson Education, Inc. Slide 5.2-2 5.2 Exponential Functions If a > 0, a  1, then f (x) = a x defines the exponential function with base a.

3. Copyright © 2011 Pearson Education, Inc. Slide 5.2-3 5.2 Graphs of Exponential Functions ExampleGraph Determine the domain and range of f. Solution There is no x-intercept. Any number to the zero power is 1, so the y-intercept is (0,1). The domain is (– ,  ), and the range is (0,  ).

4. Copyright © 2011 Pearson Education, Inc. Slide 5.2-4 5.2 Comparing Graphs ExampleExplain why the graph of is a reflection across the y-axis of the graph of Analytic Solution Show that g(x) = f (–x).

5. Copyright © 2011 Pearson Education, Inc. Slide 5.2-5 5.2 Comparing Graphs Graphical Solution The graph below indicates that g(x) is a reflection across the y-axis of f (x).

6. Copyright © 2011 Pearson Education, Inc. Slide 5.2-6 5.2 Graph of f (x) = a x, a > 1

7. Copyright © 2011 Pearson Education, Inc. Slide 5.2-7 5.2 Graph of f (x) = a x, 0 < a < 1

8. Copyright © 2011 Pearson Education, Inc. Slide 5.2-8 5.2 Using Translations to Graph an Exponential Function ExampleExplain how the graph of is obtained from the graph of Solution

9. Copyright © 2011 Pearson Education, Inc. Slide 5.2-9 5.2 Example using Graphs to Evaluate Exponential Expressions Example Use a graph to evaluate SolutionWith we find that y  2.6651441 from the graph of y = 0.5 x.

10. Copyright © 2011 Pearson Education, Inc. Slide 5.2-10 5.2 Exponential Equations (Type I) ExampleSolve Solution Write with the same base. Set exponents equal and solve.

11. Copyright © 2011 Pearson Education, Inc. Slide 5.2-11 5.2 Using a Graph to Solve Exponential Inequalities ExampleSolve the inequality SolutionUsing the graph below, the graph lies above the x-axis for values of x less than 0.5. The solution set for y > 0 is (– , 0.5).

12. Copyright © 2011 Pearson Education, Inc. Slide 5.2-12 5.2 Compound Interest Recall simple earned interest where –P is the principal (or initial investment), –r is the annual interest rate, and –t is the number of years. If A is the final balance at the end of each year, then

13. Copyright © 2011 Pearson Education, Inc. Slide 5.2-13 5.2 Compound Interest Formula ExampleSuppose that $1000 is invested at an annual rate of 4%, compounded quarterly. Find the total amount in the account after 10 years if no withdrawals are made. Solution The final balance is $1488.86. Suppose that a principal of P dollars is invested at an annual interest rate r (in decimal form), compounded n times per year. Then, the amount A accumulated after t years is given by the formula

14. Copyright © 2011 Pearson Education, Inc. Slide 5.2-14 5.2 The Natural Number e Named after Swiss mathematician Leonhard Euler e involves the expression e is an irrational number Since e is an important base, calculators are programmed to find powers of e.

15. Copyright © 2011 Pearson Education, Inc. Slide 5.2-15 5.2 Continuous Compounding Formula Example Suppose $5000 is deposited in an account paying 3% compounded continuously for 5 years. Find the total amount on deposit at the end of 5 years. Solution The final balance is $5809.17. If an amount of P dollars is deposited at a rate of interest r (in decimal form) compounded continuously for t years, then the final amount in dollars is

16. Copyright © 2011 Pearson Education, Inc. Slide 5.2-16 5.2 Modeling the Risk of Alzheimer’s Disease ExampleThe chances of dying of influenza or pneumonia increase exponentially after age 55 according to the function defined by where r is the risk (in decimal form) at age 55 and x is the number of years greater than 55. Compare the risk at age 75 with the risk at age 55. Solution x = 75 – 55 = 20, so Thus, the risk is almost fives times as great at age 75 as at age 55.