5-1 Exponential Functions (Presentation)

8/8/2019 5-1 Exponential Functions (Presentation)

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Unit 5Exponential and Logarithmic Functions

5-1 Exponential Functions 5-2 Logarithmic Functions

5-3 Exponential and Logarithmic Equations

5-4 Exponential Growth and Decay


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5-1 Exponential Functions

Unit 5 Exponential and Logarithmic Functions


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Concepts & Objectives Objective #15

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

=a

a b

b

x

( ) =b

a abx x

= ba b ax x

= =a b x x a b


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Simplifying Exponents Example: Simplify

1.

3 2 2

2 5

25

5

x y z

y z = 3 1 2 2 2 55 x y z = 2 4 35 y z

2.

3.

2 3

2r s t

i

2 3 1 25 6y y

( ) ( )

=4 2 4 34 4

2 r s t

( ) ( )

=2 1

3 25 6 y

=8 12 4

16r s t

=1

630y


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Exponential Functions Ifa > 0 and a 0, then

defines the exponential function with base a.

( ) =x

f x a

Example: Graph

Domain: (, )

Range: (0, ) y-intercept: (0, 1)

( ) = 2x

f x


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Exponential FunctionsCharacteristics of the graph of :

1. The points are on the graph.

( ) = x f x a

( ) ( )

11, , 0,1 , 1,a

a

2. Ifa > 1, thenfis an increasing function; if 0 < a < 1, thenfis a decreasing function.

3. Thex-axis is a horizontal asymptote.

4. The domain is (, ), and the range is (0, ).


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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.

Example: Solve =1

5125

x

=3

5 5x

= 3x

= 3125 5


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Exponential Equations Solve + =1 33 9x x = 29 3

( )

+

=

31 2

3 3

xx

+ =1 2 63 3x x

+ = 1 2 6x x

=7 x


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Exponential Equations To solve an equation with exponents, remember you can

undo the exponent. Solve =5 2 243b

( ) =5

243b =5 243b

or

Check your solution!

{9}

= =5 243 3b= 9b

=5 29 243

( )=

5

9 243

=53 243

=243 243

= =5 2243 59049b= =5 59049 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, where,

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.

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

t t e en o t e secon year, we ave

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

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


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Compound Interest If we continue, we end up with

Year Account

1 $1100 1000(1 + .1)

This leads us to the general formula.

+ .

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:

( )= +1t

A P r

tn

For interest compounded continuously:

= + 1 rA P

n

= 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 10.06

2500 1 $4515.28A

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

( )

= +

4 8.048

15000 14

P

( )15000 1.4648P $10,240.35P


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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?( )( )=.05 6

8000A e

10 798.87A

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?

( )

= +

12 5

.06257000 112

$9560.11

A

( )( )

=

.0575 6

7000$9883.93

A e


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Homework Algebra & Trigonometry

Page 247: 21-45 (3s) Turn in: 30, 42

College Algebra

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

5-1 Exponential Functions (Presentation)

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