Exponential Functions

• Exponential functions• Geometric Sequences

8.1 Exponential functions and their graphs

• Recognize and evaluate exponential functions with base a

• Graph exponential functions

• Recognize and evaluate exponential functions with base e

• Use exponential functions to model and solve real-life problems

Exponential functions

),0( :range ),,( :Domain number. realany is and

,0 where;)( :form thehave functions lExponentia

x

bbxf x

Graph: ( ) 4xf x ( ) 2xg x

To graph exponential functions, make a table, plot the points and connect them with a smooth curve.

x f(x) g(x)

-1

0

1

2

Domain:

Range:y -intercept:

Translations y = bx - original graph

y = abx-h + k

k - positive - moves graph upk - negative - moves graph down

h - positive - to the righth - negative - to the left

a < 0 - reflected over x - axisa > 1 - stretched0< a < 1 - compressed

Graph: ( ) 4xf x 2( ) 4xg x 24)( xxhgreen

Domain:

Range:

y -intercept

Domain:

Range:

y -intercept

A transformation of the graph:

x

y

5

2

x y

Domain: Range:

8.2 Solving exponential equations

To solve exponential equations, get the bases equal.

3 81x Solve for x: 2 12 32a

then

Rememberu va a u v

One to one property!

533 x Bases must be the same

25 1255 xx

52 2

24 xxBONUS!!

Compound Interest Formulas

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 formula:

1) For n compounding per year: 1nt

rA P

n

Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.

If $350,000 is invested at a rate of 5½% per year, find the amount of the investment at the end of 10 years for the following compounding methods:

a) Quarterly b) Monthly

Continuously Compound Interest A = Pert

Joan was born and her parents deposited $2000 into a college savings account paying 4% interest compounded continuously. What would be the balance after 15 years.

exponential decay: (1 )tA P r

• P = initial amount• (1 - r)/(1 + r) is the decay/growth factor, r is the

decay rate…0 < r < 1

• t is the time period• A = final amount

You bought a used car for $18,000. The value of the car will be less each year because of depreciation. The car depreciates (loses value) at the rate of 12% per year. Write an exponential decay model to represent the situation then use that model to estimate the value of the car in 8 years.

exponential growth:trPA )1(

Graph and find the average rate of change in value from year 0 to year 4

t y

years

value

2

2000

8.3 Logarithmic Functions

• Write exponential functions as logarithms

• Write logarithmic functions as exponential functions

A logarithm function is another way to write an exponential function

log yay x a x

where 0 and 1 and is read as

"y is the logarithm of with base of .

a a

x a

y is the logarithm, a is the base, x is the number

We can now convert from one form to the other.

273 y y 27log3

Rewrite as a exponential equation:

3log 5 c 3 5c

Rewrite as a logarithm:

2 8x 2log 8 x

To find the exact value of a logarithm (or evaluate), we can change the equation to an exponential one.

2log 16

Evaluate:

log3 81

log1/2 256

log13 169

Evaluate:

128log2

11.3/11.4 Sum of Geometric Sequences

Nth term of geometric sequence: (this is used to find any of the items or terms) an = a1rn-1

a1 = 1st termr = common ratio (divide any term by the prior term)n = how many termsWrite an equation for the nth term of a geometric sequence -.25, 2, -16, 128, ...

Find the equation for the geometric sequence: If a3 = 16, and r = 4

1

1 )( n

n raa

The geometric means are terms between non-consecutive terms of a geometric sequence.

Find the 4 geometric means between .5, __, __, __, __, 512

Partial Sum of a Geometric Series

Find the sum of the geometric seriesa1 = 2, n = 10, r = 3 r

raS

n

n

1

)1(1

Find the sum of the geometric seriesa1 = 2000, an = 125, r = 1/2

r

raaS nn

11

Find a1 in a geometric series for which Sn = -26240, n = 8, r = -3

r

raS

n

n

1

)1(1

10

3

1)2(4k

k

Or use the sum formula:

r

raS

n

n

1

)1(1

Find the sum:

12

4

1)3(4

1

k

k

r

raS

n

n

1

)1(1

r

aS

11

If , then the sequence diverges and the sum does not exist. If , the sequence converges and the sum does exist

The sum S of an infinite geometric series with

Find the sum:

...75

18

15

6

3

2