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# Exponential Functions PowerPoint PPT Presentation

Exponential Functions. The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than and x is any real number. /. Base is 2. Base is 10. Base is 3. Definition of the Exponential Function.

Exponential Functions

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### Exponential Functions

The exponential function f with base b is defined by

f (x) = bx or y = bx

Where b is a positive constant other than and x is any real number.

/

Base is 2.

Base is 10.

Base is 3.

## Definition of the Exponential Function

Here are some examples of exponential functions.

f (x) = 2xg(x) = 10xh(x) = 3x+1

f (x) = 13.49(0.967)x – 1

This is the given function.

f (31) = 13.49(0.967)31 – 1

Substitute 31 for x.

## Text Example

The exponential function f (x) = 13.49(0.967)x – 1 describes the number of O-rings expected to fail, when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, colder than any previous experience. Find the number of O-rings expected to fail at this temperature.

SolutionBecause the temperature was 31°F, substitute 31 for x and evaluate the function at 31.

f (31) = 13.49(0.967)31 – 1=3.77

f (x) = bx

0 < b < 1

f (x) = bx

b > 1

## Characteristics of Exponential Functions

• The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.

• The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.

• If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function.

• If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function.

• f (x) = bx is a one-to-one function and has an inverse that is a function.

• The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote.

Transformation

Equation

Description

Horizontal translation

g(x) = bx+c

• Shifts the graph of f (x) = bx to the left c units if c > 0.

• Shifts the graph of f (x) = bx to the right c units if c < 0.

Vertical stretching or shrinking

g(x) = c bx

• Multiplying y-coordintates of f (x) = bx by c,

• Stretches the graph of f (x) = bx if c > 1.

• Shrinks the graph of f (x) = bx if 0 < c < 1.

Reflecting

g(x) = -bx

g(x) = b-x

• Reflects the graph of f (x) = bx about the x-axis.

• Reflects the graph of f (x) = bx about the y-axis.

Vertical translation

g(x) = -bx + c

• Shifts the graph of f (x) = bx upward c units if c > 0.

• Shifts the graph of f (x) = bxdownward c units if c < 0.

g(x) = 3x+1

f (x) = 3x

(-1, 1)

(0, 1)

1

2

3

4

5

6

-5

-4

-3

-2

-1

## Text Example

Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.

SolutionExamine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs.

### Problems

Sketch a graph using transformation of the following:

1.

2.

3.

Recall the order of shifting: horizontal, reflection (horz., vert.), vertical.

f (x) = 3x

f (x) = ex

4

(1, 3)

f (x) = 2x

3

(1, e)

2

(1, 2)

(0, 1)

1

## The Natural Base e

An 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.

-1

### Formulas 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:

• For n compoundings per year:

• For continuous compounding: A = Pert.

more

more

## Example:Choosing Between Investments

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?

SolutionThe 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 compoundings per year), and t = 6.

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

## Example:Choosing Between Investments

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?

SolutionFor 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.

Use 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= PertA = 2000e(.07)(8)A = 2000 e(.56)A = 2000 * 1.75A = \$3500