3.1 Exponential Functions and Their Graphs Objectives: Students will recognize and evaluate exponential functions with base a. Students will graph exponential functions. Students will recognize, evaluate, and graph exponential functions with base e. Students will use exponential functions to model and solve real life problems.

I. Use your calculator to evaluate a. b. II. Look at the graphs of the following, compare and contrast them. a. b.

(0, 1) The exponential function has a variable as an exponent. Basic characteristics: Domain: Range: Intercept (0, 1) Increasing on x-axis is a horizontal asymptote Continuous

Natural Exponential Functions Many times, an exponential function will have a base of e. e is an irrational number. e =≈ This number is called the natural base, thus generating a natural exponential function re=related re=related

Given: Evaluate a. f( – 3 )= b. f(.41) = Between what 2 whole number bases will the graph of fall between?

Transformations of Exponential Functions a>1 vertical stretch 0<a<1 vertical compression reflection over the x-axis +c shift left –c shift right +d shift up –d shift down negative in front of x: reflection over y-axis b>1 horizontal stretch 0<b<1 horizontal compression

Describe the transformations for each of the following graphs:

Ex 1) Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. a. b. x x f(x) h(x)

Ex 2) Use the graph of f to describe the transformation that yields the graph of g. a. b.

Ex 3) (a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values.

Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas. 1.For n compoundings per year: 2.For continuous compounding:

Ex 4) Complete the table to determine the balance A for P dollars invested at rate r for t years and compounded n times per year. P = $2500r = 2.5%t = 10 years n cont. A

Work problems 65 & 67 from page 186 –187