Section 3.1

Exponential Functions

Definition

• An exponential function is in the form

• where

• and

( ) xf x a0a

1a

The Graph of an Exponential Function

• There are two cases to consider when graphing an exponential function:

• Case I: a >1

• Case II: 0 < a < 1

Case I: a > 1

• The domain is and the range is• The y-intercept is (0,1)• The graph has a horizontal asymptote y = 0

, 0,

Using Your Graphing Calculator

x

x

x

xf

xf

xf

5)(

3)(

2)(

Case II: 0 < a < 1

• The domain is and the range is• The y-intercept is (0,1)• The graph has a horizontal asymptote y = 0

, 0,

Using Your Graphing Calculator

x

x

x

xf

xf

xf

5

1)(

3

1)(

2

1)(

Transformations of Exponential Function Graphs

• All the standard rules apply.• “inside” occurs at the exponent level.• “outside” occurs at the base level.• When possible, use your graphing calculator to

help with the transformations.• When you shift a graph vertically, the horizontal

asymptote also shifts the same number of units and in the same direction.

• The range also changes as a result of a vertical shift.

Transformations of f(x) = 2x

x

x

x

x

xf

xf

xf

xf

2)(

2)(

12)(

2)( 1

The number e

• e is an irrational number

• Your scientific calculator can be used to raise e to various powers.

2.7182818e

Investing Money

• When you invest money, what are the factors that determine the return on your investment?

1. The amount you invest (P)

2. The interest rate (r)

3. The length of the investment (t)

4. The number of times per year you earn interest on the investment (n)

Two Formulas

1nt

rA P

n

rtA Pe

Compounding Table

Type of compounding n

Annually 1

Semi-annually 2

Quarterly 4

Monthly 12

Weekly 52

Daily 365

Continuously Use “pert”

Example

• Suppose we are investing $5000 for 7 years at a rate of 6%. Find the amount in the account at the end of the investment period if interest in compounded:

1.Annually

2.Quarterly

3.Weekly

4.Continuously