Exponential Functions and Their Graphs/ Compound Interest 2015/16.

Exponential Functions and Their Graphs/ Compound

Interest2015/16

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A True Story

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

http://www.youtube.com/watch?v=t3d0Y-JpRRg

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3.1 Exponential Functions and their Graphs

Objective: To use exponential functions

to solve real life problems.


4Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

The exponential function f with base a is defined by

f(x) = ax

where a > 0, a 1, and x is any real number.

For instance,

f(x) = 3x and g(x) = 0.5x

are exponential functions.


The value of f(x) = 3x when x = 2 is

f(2) = 32 =

The value of f(x) = 3x when x = –2 is

9

1

9

f(–2) = 3–2 =


The graph of f(x) = ax, a > 1

y

x(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4


The graph of f(x) = a x, a > 1

y

x

(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4

-


Example: Sketch the graph of f(x) = 2x.

x

x f(x) (x, f(x))

-2 ¼ (-2, ¼)

-1 ½ (-1, ½)

0 1 (0, 1)

1 2 (1, 2)

2 4 (2, 4)

y

2–2

2

4


Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.

x

yThe graph of this function is a vertical translation of the graph of f(x) = 2x

down one unit .

f(x) = 2x

y = –1 Domain: (–, )

Range: (–1, )

2

4


Example: Sketch the graph of g(x) = 2-x. State the domain and range.

x

yThe graph of this function is a reflection the graph of f(x) = 2x in the y-axis.

f(x) = 2x

Domain: (–, )

Range: (0, ) 2–2

4

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Example

Each of the following graphs is a transformation of

What transformation has taken place in each graph?

a) b)

c) d)


f x x( ) 3

h x x( ) 3 1 g x x( ) 3 2

k x x( ) 3 j x x( ) 3

The graph of h is the graph of f shifted one unit left.

The graph of g is the graph of f shifted down 2 units.

The graph of k is the graph of f reflected in the x-axis.

The graph of j is the graph of f reflected in the y-axis.

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Compound Interest Formulas

Interest Compounded Annually:

A = Accumulated amount after t years

P = principle (invested amount)

r = the interest rate in decimal form (5% = .05)

t = time in years

Interest Compounded n times per year:

n = # of times per year interest is compounded

(i.e. n = 1 is yearly, n = 12 is monthly, n = 365 is daily)

Continuously compounded Interest:

e is the natural number


(1 )ntrA P

n

(1 )tA P r

rtA Pee 2 718281828.

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Example: Invest $1 for 1 year at 100%Consider how n affects the accumulated amount:

Yearly n = 1

Bi-annually n = 2

Monthly n = 12

Daily n = 365

Hourly n = 8760


(1 )ntrA P

n

$2

$2.613

$2.714

$2.25

$2.718


The irrational number e, where

e 2.718281828…

is used in applications involving growth and decay.

Using techniques of calculus, it can be shown that

ne

n

n

as 1

1

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ExampleA total of $12000 is invested at 7% interest.

Find the balance after 10 years if the interest is compounded:

a) quarterly b) continuously


$24,165$24,019

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Student Example

One thousand dollars is invested in an account that earns 12% interest compounded monthly. Determine how much the investment is worth after 2 years.


$1,269.73

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Student Example

The value of a new $500 television decreases 10% per year. Find its value after 5 years.


$295.24

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Student Example

One hundred dollars is invested at 7.2% interest compounded quarterly. How much money will there be after 6 years?


$153.44

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Student ExampleLet y represent a mass of radioactive strontium, in grams, whose half-life is 28 years. The quantity of strontium present after t years is

a) What is the initial mass (when t=0)?

b) How much of the initial mass is present after 80 years?


yt10 1

228( )

10 grams

1.38 grams

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Student ExampleMrs. Johnson received a bonus equivalent to

10% of her yearly salary and has decided to deposit

it in a savings account in which interest is compounded

continuously. Her salary is $58,500 per year and

the account pays 4.5% interest.

How much interest will her deposit earn after 10 years?


$3,324.63

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Exit Ticket

A student earned $2500 working during the summer and wants to invest the money in a savings account with 7.5% interest compounded quarterly for 3 years to save for college. How much money will the student have saved for college at the end of 3 years?


$3,124.29

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HOMEWORK

3.1 pg. 185 19-27 odd, 55-59 odd, 65, 67


Exponential Functions and Their Graphs/ Compound Interest 2015/16.

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