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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