Obj. 11 Exponential Functions (Presentation)

8/3/2019 Obj. 11 Exponential Functions (Presentation)

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Obj. 11 Exponential Functions

Unit 4 Exponential and Logarithmic Functions


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Concepts & Objectives Exponential Functions (Obj. #11)

Use exponent properties to solve equations Substitute values into exponential functions


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Properties of Exponents Recall that for a variablexand integers a and b:

+

=ia b a b x x x

=

aa b

b

x

x

( ) =b

a abx x

=ba b ax x

= =a b x x a b


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Simplifying Exponents Example: Simplify

1.

2.

3.

3 2 2

2 5

25

5

x y z

y z

( )

42 3

2r s t

i2 3 1 2

5 6y y

=3 1 2 2 2 5

5 x y z

( ) ( )

=

4 2 4 34 4

2 r s t

( ) ( )

=

2 1

3 25 6 y

=2 4 3

5 y z

=8 12 4

16r s t

=

1

630y


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Exponential Functions Ifa > 0 and a 1, then

defines the exponential function with base a.

Example: Graph

Domain: ( )

Range: ( ) y-intercept: ( 1)

( ) =x

f x a

( ) = 2x

f x


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Exponential Functions

Characteristics of the graph of :

1. The points are on the graph.

2. Ifa > 1 thenf is an increasing function; if < a < 1 thenf is a decreasing function.

3. Thex-axis is a horizontal asymptote.

4. The domain is ( ) and the range is ( ).

( ) = x f x a

( ) ( )

11 1 1a

a


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Exponential Equations

Exponential equations are equations with variables as

exponents. If you can re-write each side of the equation using a

common base then you can set the exponents equal to

each other and solve for the variable.

Example: Solve =1

5125

x

=

3

5 5x

= 3x

=3

125 5


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Example: Solve+

=1 3

3 9x x

=2

9 3

( )

+

=

31 2

3 3

xx

+

=1 2 6

3 3x x

+ = 1 2 6x=7 x

( )+=

2 313 3

xx


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To solve an equation with exponents remember you can

undo the exponent by using the reciprocal. Solve

or

=5 2

243b

( ) =5

243b

= =5 243 3b= 9b

( ) ( )=2

25 552 243b

= 9b


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Compound Interest

The formula for compound interest(interest paid on

both principal and interest) is an important applicationof exponential functions.

Recall that the formula for simple interest, I= Prt, wherePis principal (amount deposited), ris annual rate of

interest, and tis time in years.


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Compound Interest Now, suppose we deposit $1000 at 10% annual interest.

At the end of the first year, we have

so our account now has 1000 + .1(1000) = $1100.

At the end of the second year, we have

so our account now has 1100 + .1(1100) = $1210.

( )( )( )= =1000 0.1 1 100I

( )( )( )= =1100 .1 1 110I


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Compound Interest If we continue, we end up with

This leads us to the general formula.

Year Account

1 $1100 1000(1 + .1)

2 $1210 1000(1 + .1)2

3 $1331 1000(1 + .1)3

4 $1464.10 1000(1 + .1)4

t 1000(1 + .1)t


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Compound Interest Formulas For interest compounded annually:

For interest compounded n times per year:

For interest compounded continuously:

where e is the irrational constant 2.718281

( )= +1t

A P r

= +

1

tn

rA Pn

=rt A Pe


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Examples1. If $2500 is deposited in an account paying 6% per year

compounded twice per year, how much is the accountworth after 10 years with no withdrawals?

2. What amount deposited today at 4.8% compoundedquarterly will give $15,000 in 8 years?

( )

= +

2 10.06

2500 12

A

( )

= +

4 8.048

15000 14

P

( )15000 1.4648P

= $10,240.35P

= $4515.28


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Examples3. If $8000 is deposited in an account paying 5% interest

compounded continuously, how much is the accountworth at the end of 6 years?

4. Which is a better deal, depositing $7000 at 6.25%compounded every month for 5 years or 5.75%compounded continuously for 6 years?

( )( )=

.05 68000A e

= $10,798.87A

( )

= +

=

12 5

.06257000 112

$9560.11

A

( )( )=

=

.0575 6

7000$9883.93

A e


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Homework College Algebra

Page 429: 51-66, (3s), 67, 68, 69-78 (3s), 83 Homework: 54, 60, 68, 72, 78

Classwork: Algebra & Trigonometry Page 247: 21-45 (3s)

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Obj. 11 Exponential Functions (Presentation)