Exponential Functions and Their Graphs Section 3.1 Exponential Functions and Their Graphs

Objective By following instructions students will be able to: Recognize and evaluate exponential functions with base a. Graph exponential functions. Recognize, evaluate, and graph exponential functions with base e. Use exponential functions to model and solve real-life problems.

Exponential Functions The exponential function f with base b is denoted by where b>0 and , and x is any real number.

Example 1: Use a calculator to evaluate each expression. a) b) c) d)

Example 2: In the same coordinate plane, sketch the graph of each function. a) b)

Example 3: In the same coordinate plane, sketch the graph of each function. a) b)

U-TRY #1 Graph. 1) 2)

Transformations of Exponential Functions Parent graph Shifts h units to the left Shift h units to the right Reflection along y axis Shift k units up Shift k units down Reflection along x axis

Example 4: Describe the transformation of a) b) c) d)

Natural base Natural base function Domain: All real numbers Range: (0, all positive real numbers) Intercept: (0,1)

Example 5: Evaluate the expression for several large values of x to see that the values approach as x increases without bound.

Example 6: Use a calculator to evaluate each expression. a) b) c) d)

Example 7: Sketch the graph of each natural function. a) b)

U-TRY #2 Graph. 1) 2)

Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (expressed as a decimal) is given by the following formulas: For n compoundings per year: For continuous compounding:

Example 8: A sum of $9000 is invested at an annual interest rate of 8.5%, compounded annually. Find the balance in the account after 3 years.

Example 9: A total of $12,000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded a. quarterly. b. monthly. c. continuously.

U-TRY #3 Complete the table to determine the balance A for P dollars invested at rate r for t years and compound n times per year. 1) 2) N 1 2 4 12 365 continuous A

Example 10: Let y represent the mass of a quantity of a radioactive element whose half-life is 25 years. After t years, the mass (in grams) is What is the initial mass when t=0? How much of the initial mass is present after 80 years?

Example 11: The approximate number of fruit flies in an experimental population after t hours is Find the number of fruit flies in the population. How large is the population of fruit flies after 72 hours? Sketch the graph of Q.

Revisit Objective Did we… Recognize and evaluate exponential functions with base a? Graph exponential functions? Recognize, evaluate, and graph exponential functions with base e? Use exponential functions to model and solve real-life problems?

Homework Pg 225 #s 1-71 EOO