1. 6.2 Exponential Functions Objective: Classify an exponential function as representing exponential growth or exponential decay. Calculate the growth of investments under various conditions. Standard: 2.11.11.C Graph and interpret rates of decay/growth

3. x y = 2 x -3 2 -3 =1/8 -2 2 -2 = ¼ -1 2 -1 = ½ 0 2 0 = 1 1 2 1 = 2 √2 2 √2 =2.67 2 2 2 = 4 3 2 3 = 8 The graph of y = 2 x approaches the x axis – but never reaches it! Notice the domain of y= 2 x includes irrational numbers, such as √2 Examine the graph of y= 2 x. Notice that as the x-values decrease, the y-values get closer and closer to 0, approaching the x-axis as an asymptote. An asymptote is a line that a graph approaches (but does not reach) as its x- or y-values become very large or very small. y= 2 x

4. The graph of f(x) = 2 x and g(x) = (1/2) x exhibit the two typical behaviors for exponential functions. g(x) = (1/2) x f(x) = 2 x g(x) = (1/2) x is a decreasing function because its base number is a positive number less than one f(x) = 2 x is an increasing exponential function because its base is a positive number greater than one

5. Geogebra Wiki

9. More Examples: y = ¼ * f(-x) y = 1/3* f(x)

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11. Ex. 2 Principal = $100 Annual Interest = 5 % Time (t) = 10

12. Effective Yield: Application Investments Suppose that you buy an item for $100 and sell the item one year later for $105. In case, the effective yield of your investments is 5%. The effective yield is the annually compounded interest rate that yields the final amount of an investment. You can determine the effective yield by fitting an exponential regression equation to two points.

13. Ex 3A. A collector buys a painting for $100,000 at the beginning of 1995 and sells it for $150,000 at the beginning of 2000. Use an exponential regression equation to find the effective yield.

14. Ex 3B. Find the effective yield for a painting bought for $100,000 at the end of 1994 and sold for $200,000 at the end of 2004.

15. 6.6 The Natural Base, e

16. The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

17. The Natural Exponential Function The exponential function with base e, f(x) = e x is called the natural exponential function and e is called the natural base. The function e x is graphed. Notice that the domain is all real numbers The range is all positive numbers.

18. Ex 1. Evaluate f(x) = e x to the nearest thousandth for each value of x below. a. x= 2 e 2 = 7.389 b. x= ½ e 1/2 = 1.649 c. x = -1 e -1 =.368 d. x = 6 e 6 = 403.429 e. x = 1/3 e 1/3 = 1.396 f. x = -2 e -2 =.135

19. Continuous Compounding Formula

20. Ex 2 An investment of $1000 earns an annual interest rate of 7.6%. Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. Quarterly A = P(1+ R/N) NT A = 1000(1+.076/4) 4*8 A = 1826.31 Continuously A = Pe rt A = 1000e.076 * 8 A = 1836.75

21. Ex 3 Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. P = 500 t = 4 R =.09 A = 500e.36 = $716.66