1. 4.2 - 1 Exponents and Properties Recall the definition of a r where r is a rational number: if then for appropriate values of m and n, For example,

2. 4.2 - 2 Exponents and Properties

3. 4.2 - 3 Exponential Function If a > 0 and a ≠ 1, then defines the exponential function with base a.

4. 4.2 - 4 Exponential Functions Slide 7 showed the graph of  (x) = 2 x with three different domains. We repeat the final graph (with real numbers as domain) here. The y-intercept is y = 2 0 = 1. Since 2 x > 0 for all x and 2 x  0 as x  – , the x-axis is a horizontal asymptote. As the graph suggests, the domain of the function is (– ,  ) and the range is (0,  ). The function is increasing on its entire domain, and is one-to- one.

5. 4.2 - 5 Domain: (– ,  ) Range: (0,  ) EXPONENTIAL FUNCTION x (x)(x) – 2– 2¼ – 1– 1 ½ 01 12 24 38   (x) = a x, a > 1, is increasing and continuous on its entire domain, (– ,  ). For  (x) = 2 x :

6. 4.2 - 6 Domain: (– ,  ) Range: (0,  ) EXPONENTIAL FUNCTION x (x)(x) – 2– 2¼ – 1– 1 ½ 01 12 24 38  The x-axis is a horizontal asymptote as x  – . For  (x) = 2 x :

7. 4.2 - 7 Domain: (– ,  ) Range: (0,  ) EXPONENTIAL FUNCTION x (x)(x) – 3– 38 – 2– 2 4 – 1– 1 2 01 1½ 2¼   (x) = a x, 0 < a < 1, is decreasing and continuous on its entire domain, (– ,  ). For  (x) = (½) x :

8. 4.2 - 8 Exponential Function The graphs of several typical exponential functions illustrate these facts.

9. 4.2 - 9 Characteristics of the Graph of  (x) = a x 1. The points and are on the graph. 2. If a > 1, then  is an increasing function; if 0 < a < 1, then  is a decreasing function. 3. The x-axis is a horizontal asymptote. 4. The domain is (– ,  ), and the range is (0,  ).

10. 4.2 - 10 Example 3 GRAPHING REFLECTIONS AND TRANSLATIONS Graph each function. Give the domain and range. (In each graph, we show the graph of y = 2 x for comparison.) Solution a. The graph of  (x) = – 2 x is that of  (x) = 2 x reflected across the x-axis. The domain is (– ,  ), and the range is (– , 0).

11. 4.2 - 11 Example 3 GRAPHING REFLECTIONS AND TRANSLATIONS Graph each function. Give the domain and range. (In each graph, we show the graph of y = 2 x for comparison.) Solution b. The graph of  (x) = 2 x+3 is the graph of  (x) = 2 x translated 3 units to the left. The domain is (– ,  ), and the range is (0,  ).

12. 4.2 - 12 Example 3 GRAPHING REFLECTIONS AND TRANSLATIONS Graph each function. Give the domain and range. (In each graph, we show the graph of y = 2 x for comparison.) Solution c. The graph of  (x) = 2 x +3 is the graph of  (x) = 2 x translated 3 units up. The domain is (– ,  ), and the range is (3,  ).

13. 4.2 - 13 Example 5 USING A PROPERTY OF EXPONENTS TO SOLVE AN EQUATION Solve Solution Write each side of the equation using a common base. Write 8 as a power of 2. Set exponents equal. Subtract 3x and 4. Divide by − 2. Check by substituting 11 for x in the original equation. The solution set is {11}.

14. 4.2 - 14 Compound Interest If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) n times per year, then after t years the account will contain A dollars, where

15. 4.2 - 15 Example 7 USING THE COMPOUND INTEREST FORMULA Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year). Find the amount in the account after 10 yr with no withdrawals. a. Solution Compound interest formula Let P = 1000, r =.04, n = 4, and t = 10.

16. 4.2 - 16 Example 7 USING THE COMPOUND INTEREST FORMULA Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year). Find the amount in the account after 10 yr with no withdrawals. a. Solution Round to the nearest cent. Thus, $1488.86 is in the account after 10 yr.

17. 4.2 - 17 Value of e To nine decimal places,

18. 4.2 - 18 Continuous Compounding If P dollars are deposited at a rate of interest r compounded continuously for t years, the compound amount in dollars on deposit is

19. 4.2 - 19 Example 10 COMPARE INTEREST EARNED AS COMPOUNDING IS MORE FREQUENT In Example 7, we found that $1000 invested at 4% compounded quarterly for 10 yr grew to $1488.86. Compare this same investment compounded annually, semiannually, monthly, daily, and continuously. Substituting into the compound interest formula and the formula for continuous compounding gives the following results for amounts of $1 and $1000. Solution

20. 4.2 - 20 Example 10 COMPARE INTEREST EARNED AS COMPOUNDING IS MORE FREQUENT Compounded $1$1000 Annually$1480.24 Semiannually $1485.95 Quarterly$1488.86 Monthly$1490.83 Daily$1491.79 Continuously$1491.82