AI I , 12 .0 : STUDENTS KNOW THE LAWS OF FRACTIONAL EXPONENTS, UNDERSTAND

EXPONENTIAL FUNCTIONS, AND USE THESE FUNCTIONS IN PROBLEMS INVOLVING EXPONENTIAL GROWTH AND DECAY.

IF YOU HAVE A CALCULATOR, USE IT !

Modeling with Exponential Functions

Objective Key Words

1. Write models for exponential growth and decay

2. Compound Interest

3. The Natural Base e

4. EC: Write an exponential function

Growth factorDecay factorNatural base

Modeling with Exponential Functions

SimplifyGreater than 1 orLess than 1

Prerequisite Check

Can you change percentages to decimals? Yes or No

Percent to Decimals

1. Now when in doubt always the number over 100:

1. Now when in doubt always the number over 100:

Exponential Growth Exponential Decay

Growth factor is r is the decimal

form of the percentage

a is the initial amount

t is the time for years

Decay factor is r is the decimal

form of the percentage

a is the initial amount

t is the time for years

Teacher Input for 1: Write Models

Example 1 Write and Use an Exponential Growth Model

Sales Figures One year, a clothing company had $1.5 million in sales. In later years, sales y (in millions of dollars) increased by about 25% each year.

a. Write an exponential growth model that represents the sales after t years.

b. Use the model to predict the sales after 8 years.

SOLUTION

a. =y a ( )tr+1 Write exponential growth model.

= 1.5( )t0.25+1 Substitute 1.5 for a and 0.25 for r.

= 1.5( Simplify.)t1.25

Example 1 Write and Use an Exponential Growth Model

ANSWER The model is = 1.5( )t.1.25y

b. To predict the sales after 8 years, substitute 8 for t.

y = 1.5( )81.25 ≈ 8.9

ANSWER

The sales after 8 years will be about $8.9 million.

You Try!!!

1. Redo Example 1 using a sales increase of 10% each year.

Write and Use an Exponential Growth Model

ANSWER

= 1.5( )t1.1ya.

b. $3.2 million

Example 2 Write and Use an Exponential Decay Model

Computers You buy a new computer for $1500. The value y (in dollars) of the computer decreases by 40% each year.

SOLUTION

a. Let t be the number of years since you bought the computer.

a. Write an exponential decay model that represents the value of the computer.

b. Use the model from part (a) to estimate the value after 3 years.

ay = ( )t1 – r Write exponential decay model.

Example 2 Write and Use an Exponential Decay Model

1500= ( )t1 – 0.4 Substitute 1500 for a and 0.4 for r.

1500= ( Simplify.)t0.6

ANSWER The model is 1500y = ( )t.0.6

b. To estimate the value after 3 years, substitute 3 for t.

1500= ( )30.6y 324=

ANSWER

The value of the computer after 3 years is $324.

You Try!!!

2. Redo Example 2 using a new computer cost of $1200 and a value decrease of 35% each year.

Write and Use an Exponential Decay Model

ANSWER

a. 1200y = ( )t0.65

b. $329.55

Teacher input for 2: Compound Interest

An initial principal P is deposited in an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year.

The amount A is the account after t years can be modeled by this formula:

Example 3 Find the Balance in an Account

Finance You deposit $2000 in an account that pays 2% annual interest. Find the balance after 10 years if the interest is compounded quarterly.

Write compound interest formula.

=Ant

n

r

SOLUTION

1 +P

Substitute 2000 for P, 0.02 for r, 4 for n, and 10 for t.

=4 10

4

0.021 +2000

•

Simplify.= 2000 1.005( )40

Use a calculator.2441.59≈

Example 3 Find the Balance in an Account

ANSWER The balance after 10 years is about $2441.59.

Using the graph, you can see that after 10 years, the balance isbetween $2400 and $2500.

CHECK You can check the solution by graphing the function A = 2000 1.005( )4t.

You Try!!!

6. You deposit $1500 in an account that pays 2% annual

interest. Find the balance after 6 years if the interest is compounded monthly.

Write and Use Exponential Functions

ANSWER $1691.08

Teacher input 3: Natural Base e

The natural base e is irrational. (Euler number)

It is defined as follows: As n approaches +∞,

approaches

Compound Interest Formula

Continuously Compounded Interest Formula

Example 4 Find the Balance in an Account

Finance You deposit $500 in an account that pays 4% annual interest compounded continuously.

a. Find the balance after one year.

b. Graph the continuously compounded interest model.

c. Use the graph to estimate how long it will take your money to grow to $700.

Substitute 500 for P, 0.04 for r, and 1 for t.= 500e 0.04 1•

SOLUTION

Write continuously compounded interest formula.

=A Pe rta.

Example 4 Find the Balance in an Account

Simplify.= 500e 0.04

Use a calculator.520.41≈

ANSWER The balance after one year is about $520.41.

c. Use the Trace feature to determine when the balance is $700. This happens when x is about 8.4. So, it will take about 8.4 years for your money to grow to $700.

b. The graph of the model is shown at the right.

=y 500e 0.04x

You Try!!!

7. You deposit $1000 in an account that pays 3% annual interest compounded continuously. Find the balance after 1 year, 3 years, and 5 years.

Find the Balance in an Account

ANSWER $1030.45, $1094.17, $1161.83

Summary Assignment

How can you write a model for an exponential function? The form of an

exponential function is , where a and b are constants. You can use information in the problems to find the values of a and b.

Pg 430 #(14-25,29,30 ALL)

EC Pg 431 #(33-43 ODD)

Problems not finished in class are left as homework

Conclusions

JUST AS TWO POINTS DETERMINE A LINE, TWO POINTS ALSO DETERMINE AN

EXPONENTIAL CURVE

Write an Exponential Function

Example 5 Write an Exponential Function

=y ab x

1, 6( ) 2, 18( )

SOLUTION

Substitute the coordinates of the points into to obtain two equations.

=y ab x

Substitute 6 for y and 1 for x, because (1, 6) is on the graph.

=6 ab 1

Substitute 18 for y and 2 for x, because (2, 18) is on the graph.

=18 ab 2

Solve the first equation for a to get . Then

substitute into the second equation.

=ab

6

Write a function of the form whose graph passes through and .

Example 5 Write an Exponential Function

Substitute for a.=18 b 2

b

6

b

6

Quotient of powers property=18 6b

Divide each side by 6.=3 b

ANSWER

Using , you find that Because

and , is the exponential function whose

graph passes through and

=b 3 =ab

6=

3

62.= =a 2

=b 3 =y 3x2 •

1, 6( ) 2, 18( ).

You Try!!!

3.

Write an exponential function of the form whose graph passes through the given points.

Write and Use Exponential Functions

=y ab x

2, 16( ), 3, 64( ) ANSWER =y 4x

4. 2, 3( ), 4, 12( ) ANSWER =y 2x•4

3

5. 1, 3( ), 3, 108( ) ANSWER =y 6x•2

1

Practice Work Assignment

Complete in class: Pg459 #(ALL)

Ask others for helpFor teacher help you

must provide 3 names of those you asked

All 4 of those unable to help will be given help

Pg456 #(13-16 ALL)

Problems not finished in class are left as homework

Conclusion