Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2

Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE

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Compare the graphs 2x, 3x , and 4x

Characteristics about the Graph of an Exponential Function where a > 1 xaxf

What is the domain of an exponential function?

1. Domain is all real numbers

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What is the range of an exponential function?

2. Range is positive real numbers

What is the x intercept of these exponential functions?

3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

What is the y intercept of these exponential functions?

4. The y intercept is always (0,1) because a 0 = 1

5. The graph is always increasing

Are these exponential functions increasing or decreasing?

6. The x-axis (where y = 0) is a horizontal asymptote for x -

Can you see the horizontal asymptote for these functions?

All of the transformations that you learned apply to all functions, so what would the graph of look like?

xy 232 xy

up 3

xy 21up 1

Reflected over x axis 12 2 xy

down 1right 2

xy 2

Reflected about y-axis This equation could be rewritten in a different form: x

xxy

2

1

2

12

So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.

There are many occurrences in nature that can be modeled with an exponential function. To model these we need to learn about a special base.

The Natural Base : e

Instead of using base as a number, in application problems, we can use base e.

•e : Natural base

•e = 2.718281828…..

•You need to remember this value

The Base “e” (also called the natural base)

To model things in nature, we’ll need a base that turns out to be between 2 and 3. Your calculator knows this base. Ask your calculator to find e1. You do this by using the ex button (generally you’ll need to hit the 2nd or yellow button first to get it depending on the calculator). After hitting the ex, you then enter the exponent you want (in this case 1) and push = or enter. If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex. You should get 2.718281828

Example for TI-83

• Well, let me show you how to remember:

• Remember to start with 2.

• Who was the 7th president of US?

• ANDREW JACKSON

• When was he elected?

• Make a square with sides 1828.

• Write 1828 twice.

• Make a diagonal. What type of triangle is this?

• And so on………I am tired.

• 2.718281828459045…….

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xxf 3

xexf

The Natural Base eAn irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately,

The number e is called the natural base. The function f (x) = ex is called the natural exponential function.

2.71828...e

-1

Translations:

Application:Application:Formulas for Compound InterestFormulas for Compound Interest

After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1. For n compounding per year:

2. For continuous compounding: A = Pert.

1nt

rA P

n

Example: Choosing Between InvestmentsYou want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?

Solution The better investment is the one with the greater balance in the account after 6 years. Let’s begin with the account with monthly compounding. We use the compound interest model with P = 8000, r = 7% = 0.07, n = 12 (monthly compounding, means 12 compounding's per year), and t = 6.

The balance in this account after 6 years is $12,160.84.

12*60.071 8000 1 12,160.8412

ntrA Pn

moremore

You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?

Solution For the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = 0.0685, and t = 6.

The balance in this account after 6 years is $12,066.60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option.

0.0685(6)8000 12,066.60rtA Pe e

ExampleUse A= Pert to solve the following problem:

Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously

Solution:A= Pert

A = 2000e(.07)(8)

A = 2000 e(.56)

A = 2000 * 1.75A = $3500

Try: