A genie offers you a choice: He will give you $50,000 right now OR He will give you 1 penny today, 2 tomorrow, 4 the next day and so on for a month. Which do you choose?

3.1 Exponential Functions and Their Graphs  Students will be able to:  Evaluate exponential functions  Graph exponential functions and their transformations

Definition of Exponential Functions The exponential function f with base a is denoted by: f(x) = a x, where a > 0, a ≠1, and x is any real number  Exponential functions increase or decrease very quickly  If a>1, the function will be increasing (exponential growth)  If a<1, the function will be decreasing (exponential decay)

Evaluating Exponential Functions  Use a calculator to evaluate each function at the indicated value of x. FunctionValue a)f(x) = 2 x x = -3.1 b)f(x) = 2 -x x = π c)f(x) =.6 x x = 3/2

Graphing Exponential Functions xf(x)g(x) “Well that escalated quickly!”

 pg 185 #2-5 all, 7-13 odd, all  Finish orange packet  It is a well known fact that a normal piece of paper cannot be folded more than 7 or 8 times, but imagine it could be folded 42 times– how tall do you think it would be? Homework and a Riddle

Properties of Exponential Functions f(x) = a x, a > 0, a ≠ 1  Domain:  Range:  x-intercept:  y-intercept:  Inc/Dec:  Continuous:  HA:

Transformations of Exponential Functions TransformationGraphical ChangeMathematical Change c = -1 Reflected over y-axisBecomes the reciprocal (x, y)  (-x, y) b = -1 Reflected over x-axisy values are negative (x, y)  (x, -y) b Stretch the graph verticallyMultiply y values by b (x, y)  (x, by) c Stretch the graph horizontally d Move the graph to the left (+) or right (-)(x, y)  (x–d, y) e Move the graph up (+) or down (-)(x, y)  (x, y+e)

Solving Exponential Equations Using the 1-1 Property

The natural base: e

Evaluating the Natural Exponential Function Use a calculator to evaluate the function f(x) = e x a) x = -2 b) x =.25 c) x = -.4

Graphing Natural Exponential Functions  f(x) = 2e.24x xf(x)

 Many things that grow or shrink over time are best modeled with exponential growth or decay functions.  Interest rates, population growth, car depreciation, and carbon dating are all common examples.  The general formula for an exponential function is Word Problems

 How thick would it be if you could fold it 42 times? Write a function to determine how thick a piece of paper would be if you folded it in half x times. A piece of paper is inches thick.

Interest Problems Simple Interest  Interest always calculated on the principal amount  Linear function Compound Interest  Interested is calculated based on the current balance, taking into consideration previously earned interest  Exponential function

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.

The annual interest rate on a $12,000 loan is 3.8%. Find how much is owed after 4 years if the interest is compounded (a) quarterly and (b) continuously.  (a)  (b)

Determine the amount of money that should be invested at 9% interest, compounded monthly, to produce a final balance of $30,000 in 15 years.

 Pg 186 #55, 58, 64, 65, 68, 70  #51, 53, 54, 57 below Homework