Download Exponential Functions

Warm Up
1. ( x
2.
3
4
Jan. 4th
2
8
y )( x y z )
(3a b ) (2a b c)
3 4 2
3
3.  3 x y 
 18 x  2 y 7 


2 0 6 3
4. r p s r
3
3
4
5. 16 x y
3
28 yx
2 5
Exponential Functions
An exponential function is a function
of the form y  a  b ,
where a  0, b  0, and b  1,
and the exponent must be a variable.
x
The can be described by
having a constant rate of
change.

However, unlike linear their rate of
change is a constant rate of
multiplication or division.
y  a b
x
So our general form is simple enough.
The general shape of our graph will be determined
by the exponential variable.
Which leads us to ask what role does the ‘a’ and the
base ‘b’ play here. Let’s take a look.

Our general exponential form is y  a  b

“b” is the base of the function and
changes here will result in:


When b>1, a steep increase in the value of
‘y’ as ‘x’ increases.
When 0<b<1, a steep decrease in the value
of ‘y’ as ‘x’ increases.
x
Review: what is an
asymptote?
“Walking halfway to the wall”
y  10(2)
In this
An Asymptote example,
is a line that a
the
graph
asymptote
approaches as is the x
x or y
axis.
increases in
absolute
value.
x
Graphing

Ex: Graph
x
y=ab
when a<0
1 x
1 x
y  (2) and y   (2)
2
2
Sketch your
prediction of
what the graph
will look like
Where is the
asymptote?
Translating

y
x
y=ab
How does the equation change if we want to
move both graphs up 4 units? Predictions?
1 x
1
(2) and y   (2) x
2
2
1 x
1 x
y  (2)  4 and y   (2)  4
2
2
Question:
where is the
asymptote
now? To move the graph up or down, add or subtract
units at the end of the equations. No need to use
inverses – if you want to go up, add; if you want
to go down, subtract.
Translating

x
y=ab
How does the equation change if we want to
move both graphs left 4 units? Predictions?
1 x
1 x
y  (2) and y   (2)
2
2
1
1
y  (2) x  4 and y   (2) x  4
2
2
To move the graph left or right, add or subtract units to the
exponent. Reminder: use the inverse of how you want the graph
to move (e.g. x-4 moves to the right; x+4 moves to the left)
Let’s try some

What transformation
is occurring in the
function if the
parent graph is
y=9(3)x
a) y  9(3)
x 1
b) y  9(3)  4
x
c) y  9(3)
x4
1
“e = 2.718”
What is base “e” ?
e is an irrational number, approximately equal to 2.718.
Exponential
functions with a base
of e are useful for
describing
continuous growth or
decay. In the graph
below, y = e is the
asymptote to the
graph.
y=e
Graphing
x
e

Using your graphing calculators,
graph y=ex. Evaluate e4 to four
decimal places.
We now
need to
evaluat
e
where
x=4
2. Press 2nd, Calc
and select 1
(value). Press
enter
3. We are
evaluating when
x=4. Enter 4 for x
and press enter.
The value of e4 is
about 54.59815
Your turn: evaluate e-3
0.0498
So, what is “e” good for???
Evaluating an Exponential
Function
1. Example: Suppose 30 flour beetles are
left undisturbed in a warehouse bin. The
beetle population doubles each week.
How many beetles will there be after 56
days?
Step 1: Create Function.
Step 2: Convert 56 days to weeks.
Example:

a)
BUSINESS The amount of money spent at West
Outlet Mall in Midtown continues to increase. The
total T(x) in millions of dollars can be estimated by
the function T(x)=12(1.12)x, where x is the number
of years after it opened in 1995.
According to the function, find the amount of sales
in 2006, 2008 and 2010.
b)
Name the y-intercept.
c)
What does it represent in this problem?
You Try
3. An initial population of 20 rabbits
triples every half year. The function
gives the population after x half-year
periods. How many rabbits will there
be after 3 years?
Continuously Compounding
Interest

A = Pert
A = amount of money in
the account
P = principal (how much
is deposited)
r = annual rate of interest
t = time (in years)
Example: Continuously
Compounded Interest Problem

You invest $1,050 at an annual interest rate of 5.5%,
compounded continuously. How much will you have in
the account after 5 years?
•Start with: P=$1050, r=5.5% = 0.055, t=5
A = Pert
Let’s try one:

Suppose you invest $1,300 at an
annual interest rate of 4.3%,
compounded continuously. How much
will you have in the account after three
years?
Suppose you invest $1,300 at an annual interest rate of
4.3%, compounded continuously. How much will you
have in the account after three years?
We know that in exponential functions the
exponent is a variable.
The Equality Property for Exponential
Functions
Suppose b is a positive number other
than 1. Then b x  b x
1
2
if and only if
x1  x 2 .
Basically, this states that if the bases are the same, then we
can simply set the exponents equal.
This property is quite useful when we
are trying to solve equations
involving exponential functions.
Let’s try a few examples to see how it works.
Example 1:
2x5
3
3
x 3
2x  5  x  3
x5 3
(Since the bases are the same we
simply set the exponents equal.)
x8
Here is another example for you to try:
Example 1a:
3x 1
2
2
1
x 5
3
The next problem is what to do when the bases are
not the same.
2x  3
3
 27
x1
Does anyone remember how
we might approach this?
Our strategy here is to rewrite the bases so that they
are both the same.
Here for example, we know that
3  27
3
Example 2: (Let’s solve it now)
2x  3
3
 27
x1
Example 3
16
x 1
1

32
By now you can see that the equality property is
actually quite useful in solving these problems.
Here are a few more examples for you to try.
Example 4:
Example 5:
2x 1
3
4
x 3
1

9
8
2x 1