1. Exponential Functions

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

3. Examples What are the domain and range of y = 2(3 x ) – 4? What are the roots of 0 =5 – 2.5 x ?

4. Properties of Powers (Review) When multiplying like bases, add exponents. a x ● a y = a x+y When dividing like bases, subtract exponents. When raising a power to a power, multiply exponents. (a x ) y =a xy

5. 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

6. 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 = ka x, if a > 1  k is the initial amount present  a is the rate at which the amount is growing  Exponential Decay: y = ka x, 0 < a < 1  k is the initial amount present  a is the rate at which the amount is growing

7. 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: After 40 days: IN GENERAL: Models the mass of the substance after t days. Therefore, let graph, and find intersection. t ≈ 46.44 days

8. 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?

9. Exponential Growth Example: How long will it take a population to triple if it is increasing at a rate of 2.75%?

10. The Number e Many real-life phenomena are best modeled using the number e ▫e ≈ 2.718281828 e can be approximated by: Interest compounding continuously: I = Pe rt, where P = initial investment, r = interest rate (decimal) t = time in years

11. 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

12. Suggested HW Sec. 1.3 (#5, 7, 11, 19, 21-31 odd) 1.3 Web Assign Due Monday night