5.2 exponential functions

Quiz Fill in the blanks below: 2x+y = __ * __

Exponential function Standard form: f(x) = ax , where a>0, a ≠ 1. Example: f(x) = 2x, f(x) = (1/3)x Compare g(x) = x2 and f(x) = 2x

properties Graph f(x) = 2x and g(x) = (1/2)x

Properties Characteristics of f(x) Continuous One to one Domain: (- ∞, ∞) Range: (0, ∞) Increasing if a>1(growth) Decreasing if 0<a<1(Decay) Horizontal asymptote at y = 0. Key points on the graph: (1,a), (0,1)

Graphs by transformations Describe how each of the following can be obtained from the graph of f(x) = 2x. a. f(x) = 2x+3 b. f(x) = 2x – 1 c. f(x) = 3 + 2-x Example: exercise #37, #69

Exponential Equations ab = ac  b = c Solve for x: 1. 2x-3 = 8 2. (1/4)3 = 8x 3. 274x = 9x+1 Use a graphing calculator to solve 2x-3 > 8 or 2x-3 ≤ 8

Natural Base -- e e = (1 + 1/k)k as k approaches positive infinite. Natural exponential function: f(x) = ex

A = P(1 + r/n)nt Compound interest Suppose that a principal of P dollars is invested at an annual interest rate r, compounded n times per year. Then the amount A accumulated after t years is given by the formula A = P(1 + r/n)nt A = Amount accumulated after t years P = principal r = annual interest rate n = compounded times of a year

Typical Compounding periods Compound annually: n = 1 Compound semi- annually: n = 2 Compound quarterly: n = 4 Compound monthly: n = 12 Compound weekly: n = 52 Compound daily: n = 365

example Suppose that $100,000 is invested at 6.5% interest, compounded semi-annually. 1. Find a function for the amount of money after t years 2. Find the amount of money in the account at t = 1,4,10 years.

Continuous Compounding As the number of compounding periods increases without bound, the model becomes A = Pert

example If you put $7000 in an money market account that pays 4.3% a year compounded continuously, how much will be in the account in 15 years? You have $1500 to invest. Which is better – 2.25% compounded quarterly for 3 years? Or 1.75% compounded continuously for r years?

Homework PG. 339: 3-18(M3), 38, 39-75(M3), 80 KEY: 38, 54, 60, 75 Reading: 5.3 Logarithms and their properties