Exponential Functions Exponential Function fx ax for any

Exponential Functions

Exponential Function f(x) = ax for any positive number a other than one.

Examples • What are the domain and range of y = 2(3 x) – 4? • What are the roots of 0 =5 – 2. 5 x?

Properties of Powers (Review) • When multiplying like bases, add exponents. ax ● ay = ax+y • When dividing like bases, subtract exponents. • When raising a power to a power, multiply exponents. (ax)y=axy

Properties of Powers (Review) • When you have a monomial or a fraction raised to a power (with no add. or sub. ), raise everything to that power. or

Half-Life & Exponential Growth/Decay • The half-life of a substance is the time it takes for half of a substance to exist. ▫ Mirrors the behavior of Exponential Growth & Decay functions. �Exponential Growth: y = kax, if a > 1 �k is the initial amount present �a is the rate at which the amount is growing �Exponential Decay: y = kax, 0 < a < 1 �k is the initial amount present �a is the rate at which the amount is growing

Example • Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining? After 20 days: IN GENERAL: Models the mass of the substance after t days. After 40 days: Therefore, let graph, and find intersection. t ≈ 46. 44 days

Exponential Growth/Decay Example: A population initially contains 56. 5 grams of a substance. If it is increasing at a rate of 15% per week, approximately how many weeks will it take for the population to reach 281. 4 grams?

Exponential Growth Example: How long will it take a population to triple if it is increasing at a rate of 2. 75%?

The Number e • Many real-life phenomena are best modeled using the number e ▫ e ≈ 2. 71828 • e can be approximated by: • Interest compounding continuously: I = Pert, where P = initial investment, r = interest rate (decimal) t = time in years

Example Compounding Interest • A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounded a. ) quarterly b. ) monthly c. ) continuously

Suggested HW • Sec. 1. 3 (#5, 7, 11, 19, 21 -31 odd) • 1. 3 Web Assign Due Monday night