Exponential Functions and Their Graphs Skill 16

Objectives… Recognize and evaluate exponential functions with base a. Graph exponential functions with base a. Recognize, evaluate, and graph exponential functions with base e. Use exponential functions to model and solve real-life problems.

Example–Evaluating Exponential Functions Evaluate each function at the indicated value of x. Function Value a. f (x) = 2x x = –3.1 b. f (x) = 2–x x =  c. f (x) = 0.6x x = 3 2 d. f (x) = 1.052x x = 12

Example–Solution

Example–Graphs of y = ax In the same coordinate plane, sketch the graph of each function by hand. a. f (x) = 2x b. g (x) = 4x

Graphs of Exponential Functions The parent exponential function f (x) = ax , a > 0, a  1 Many real-life phenomena with patterns of rapid growth (or decline) can be modeled by exponential functions.

Graphs of Exponential Functions x-axis is a horizontal asymptote (ax  0 as x  -∞) Continuous x-axis is a horizontal asymptote (a–x  0 as x  ∞) Continuous

The Natural Base e For many applications, the convenient choice for a base is the irrational number e = 2.718281828 . . . . This number is called the natural base. The function f (x) = ex is called the natural exponential function. The Natural Exponential Function

Example–Evaluating the Natural Exponential Functions Evaluate the function at each indicated value of x f (x) = ex a. x = –2 b. x = 0.25 c. x = –0.4 d. x = 2 3

Example–Solution

Applications One of the most familiar examples of exponential growth is an investment earning continuously compounded interest.

Applications

Example–Finding the Balance for Compound Interest A total of $9000 is invested at an annual interest rate of 2.5%, compounded annually. Find the balance in the account after 5 years. Solution: In this case, P = 9000, r =2.5% = 0.025, n = 1, t = 5. Using the formula for compound interest Formula for compound interest

Example– Solution = 9000(1.025)5  $10,182.67. So, the balance in the account after 5 years will be about $10,182.67. Substitute for P, r, n, and t. Simplify. Use a calculator.

Example–Finding the Balance for Compound Interest A total of $13,000 is invested at an annual interest rate of 3.5%, compounded quarterly. Find the balance in the account after 3 years. Solution: In this case, P = 13,000, r =3.5% = 0.035, n = 4, t = 3. Using the formula for compound interest Formula for compound interest

Example– Solution 13,000 1+ .035 4 4(3) = 13,000(1.00875)12 13,000 1+ .035 4 4(3) = 13,000(1.00875)12  $14,432.64 So, the balance in the account after 3 years will be about $14,432.64. Substitute for P, r, n, and t. Simplify. Use a calculator.

16 Exponential Functions and Their Graphs Summarize Notes Questions? Homework Worksheet

Graphs of Exponential Functions… Graph of f (x) = ax , a > 1 Graph of f (x) = a –x , a > 1 Domain:( , ) Domain:( , ) Range :(0 , ) Range :(0 , ) Intercept :(0 ,1) Intercept :(0 ,1) Increasing on :( , ) Increasing on :( , )