*Stacie Bender 9/15/14*

An exponential function is a function in which the variable is an exponent. What would that look like? When we say something is a function, we begin with function notation:

f(x) =

Since the variable must be the exponent, we need a constant or a number to the power of x.

f(x) = a^{x}

So f(x) = 7 ^{x}, f(x) = ½ ^{x}, and f(x) = .4^{x} are all exponential functions.

Is f(x) = x ^{2} an exponential function? No, the exponent has to be the variable.

Exponential functions are defined where a > 0 and x is any number.

Looking at this graph of f(x) = a^{x}, what is the domain?

Looking at the graph of f(x) = a^{-x}, what is the domain?

What is the range?

Where is it increasing?

Decreasing?

What is the y-intercept?

## Graph an Exponential Function

Graph f(x) = 2^{x}

## Families of Graphs

f(x) = a (x – h) + k

a – stretch (amplitude)

h – horizontal movement

k – verti*k*al** **movement

If f(x) = 2^{x}, describe the transformation when g(x) = 2^{x + 3}

Describe the transformation when h(x) = 2^{-x} – 4.

## e – Euler’s Number, Napier’s Constant, the Natural Base

e is an irrational number that’s important in Calculus limits, continuously compounded interest, and exponential decay. It is an irrational number.

e ≈ 2.718281

e is the base in the natural log noted as ln.

So ln = log_{e} ≈ log _{2.718}

**Compound n times/year Continuously Compound**

A =

P =

n =

r =

t =

** **

If you loaned a bank $10,000 at a 5% interest rate for 10 years, find out the amount you would have in the bank if it is compounded:

Monthly

Continuously

Precalculus with Limits – Larson & Hostetler p 218 – 228