MAT 150 AlgebraClass #17

Objectives

Graph and apply exponential functionsFind horizontal asymptotesGraph and apply exponential growth functionsGraph and apply exponential decay functionsCompare transformations of graphs of exponential functions

Exponential FunctionIf b is a positive real number, b 1, then the functionf(x) = bx is an exponential function. The constant b is called the base of the function, and the variable x is theexponent.

Example

Explain how the graph of each of the following functions compares with the graph of y = 2x, and graph each function on the same axes as y = 2x.a.Solution

32xy

Example

Explain how the graph of each of the following functions compares with the graph of y = 2x, and graph each function on the same axes as y = 2x.b.Solution

33 2xy

Example

Explain how the graph of each of the following functions compares with the graph of y = 2x, and graph each function on the same axes as y = 2x.c.Solution

4(2 )xy

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(1.042) the following year.a. Write the function that gives the cost of a $10,000 item

t years after 2012.Solution

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(1.042)the following year.b. Graph the growth model found in part (a) for t = 0 to t = 13. Solution

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(1.042) the following year.c. If an item costs $10,000 in 2012, use the model to

predict its cost in 2025. Solution

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(20.5x ) dollarswhere 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?

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

Example

c. Use a graph of the function to estimate the week in which sales equal $500.

Solution

Example

d. According to this model, will sales ever fall to zero?

Solution

The Number e

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

Example

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

Solution

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