1. MAT 150 Algebra Class #17

2. Objectives  Graph and apply exponential functions  Find horizontal asymptotes  Graph and apply exponential growth functions  Graph and apply exponential decay functions  Compare transformations of graphs of exponential functions

3. Exponential Function If b is a positive real number, b  1, then the function f(x) = b x is an exponential function. The constant b is called the base of the function, and the variable x is the exponent.

5. Example Explain how the graph of each of the following functions compares with the graph of y = 2 x, and graph each function on the same axes as y = 2 x. a. Solution The graph has the same shape. Shifted 3 units right.

6. Example Explain how the graph of each of the following functions compares with the graph of y = 2 x, and graph each function on the same axes as y = 2 x. b. Solution The graph has the same shape. Shifted 3 units right and 3 units up. The horizontal asymptote is y = 3.

7. Example Explain how the graph of each of the following functions compares with the graph of y = 2 x, and graph each function on the same axes as y = 2 x. c. Solution The graph stretches by a factor equal to the constant 4.

9. Example Suppose that inflation is predicted to average 4% per year for each year from 2012 to 2025. This means that an item that costs $10,000 one year will cost $10,000(1.04) the next year and $10,000(1.04)(1.04) = 10,000(1.04 2 ) the following year. a. Write the function that gives the cost of a $10,000 item t years after 2012. Solution

10. Example Suppose that inflation is predicted to average 4% per year for each year from 2012 to 2025. This means that an item that costs $10,000 one year will cost $10,000(1.04) the next year and $10,000(1.04)(1.04) = 10,000(1.04 2 ) the following year. b. Graph the growth model found in part (a) for t = 0 to t = 13. Solution Graph is shown

11. Example Suppose that inflation is predicted to average 4% per year for each year from 2012 to 2025. This means that an item that costs $10,000 one year will cost $10,000(1.04) the next year and $10,000(1.04)(1.04) = 10,000(1.04 2 ) the following year. c. If an item costs $10,000 in 2012, use the model to predict its cost in 2025. Solution The year 2025 is 13 years from 2012.

13. Example It pays to advertise, and it is frequently true that weekly sales will drop rapidly for many products after an advertising campaign ends. This decline in sales is called sales decay. Suppose that the decay in the sales of a product is given by S = 1000(2  0.5x ) dollars where x is the number of weeks after the end of a sales campaign. Use this function to answer the following. a. What is the level of sales when the advertising campaign ends? b. What is the level of sales 1 week after the end of the campaign? c. Use a graph of the function to estimate the week in which sales equal $500. d. According to this model, will sales ever fall to zero?

14. Example a. What is the level of sales when the advertising campaign ends? b. What is the level of sales 1 week after the end of the campaign? Solution a. The campaign ends when x = 0, so b. End of week 1, x = 1, so

15. Example c. Use a graph of the function to estimate the week in which sales equal $500. Solution c. One way to find for which S = 500 is to graph the two functions and find the point of intersection. y = 500 when x = 2 Thus, sales fall to half their original amount after 2 weeks.

16. Example d. According to this model, will sales ever fall to zero? Solution d. The graph of the sales decay function approaches the positive x-axis as x gets large, but it never reaches the x-axis. The sales will never reach a value of $0.

17. The Number e The number e is an irrational number with a decimal approximation of 2.718281828.

18. Example If $10,000 is invested for 15 years at 12% compounded continuously, what is the future value of the investment? Solution

19. Assignment Pg. 321-323 #1-3 #13-18 #23-25*(Graph both functions on the same graph and then compare the two. You need to be detailed when graphing.) #29,33,40