1. THE FOLLOWING PRESENTATION
COLECTED FROM DIFFERENT
SLIDESHARE
DATE ACCESSED: 03 MARCH 2014

2. Exponential Functions
More Mathematical Modeling

3. Internet Traffic
 In 1994, a mere 3 million
people were connected to
the Internet.
 By the end of 1997, more
than 100 million were
using it.
 Traffic on the Internet
has doubled every 100
days.
 Source: The Emerging
Digital Economy,
April 1998 report of the
United States Department of
Commerce.

4. Derivatives of
Exponential Functions

5. Exponential Functions
 A function is called an exponential
function if it has a constant growth
factor.
 This means that for a fixed change in x,
y gets multiplied by a fixed amount.
 Example: Money accumulating in a bank
at a fixed rate of interest increases
exponentially.

6. Exponential Function
 An exponential equation is an equation in which the
variable appears in an exponent.
 Exponential functions are functions where f(x) =
ax + B,
where a is any real constant and B is any expression.
For example,
f(x) = e-x - 1 is an exponential function.
 Exponential Function:
f(x) = bx or y = bx,
where b > 0 and b ≠ 1 and x is in R
For example,
f(x) = 2x
g(x) = 10x
h(x) = 5x+1

7. Exponential Equations with Like Bases
 Example #1 - One exponential expression.
 Example #2 - Two exponential expressions.
Evaluating Exponential Function
32x 1
5 4
32x 1
9
32x 1
32
2x 1 2
2x 1
x
1
2
1. Isolate the exponential
expression and rewrite the
constant in terms of the same
base.
2. Set the exponents equal to
each other (drop the bases) and
solve the resulting equation.
3x 1
9x 2
3x 1
32 x 2
3x 1
32x 4
x 1 2x 4
x 5

8. Exponential Equations with Different Bases
 The Exponential Equations below contain exponential
expressions whose bases cannot be rewritten as the same
rational number.
 The solutions are irrational numbers, we will need to use a log
function to evaluate them.
 Example #1 - One exponential expression.
32x 1
5 11 or 3x 1
4x 2
32 x 1
5 11
32 x 1
16
ln 32x 1
ln 16
(2x 1)ln3 ln16
1. Isolate the exponential expression.
3. Use the log rule that lets you rewrite
the exponent as a multiplier.
2. Take the log (log or ln) of both sides
of the equation.

9. Exponential Functions
 Consider the following example, is this
exponential?
x y
5 0.5
10 1.5
15 4.5
20 13.5

10. Exponential Functions
 For a fixed change in x, y gets multiplied by a
fixed amount. If the column is constant, then
the relationship is exponential.
x y
5 0.5
10 1.5 1.5 / 0.5 3
15 4.5 4.5 / 1.5 3
20 13.5 13.5 / 4.5 3

11. This says that if we have exponential functions in
equations and we can write both sides of the equation
using the same base, we know the exponents are equal.
If au = av, then u = v
82 43x The left hand side is 2 to the something.
Can we re-write the right hand side as 2
to the something?
343
22 x
Now we use the property above. The
bases are both 2 so the exponents must
be equal.
343x We did not cancel the 2’s, We just used
the property and equated the exponents.
You could solve this for x now.

12. Let’s examine exponential functions. They are
different than any of the other types of functions we’ve
studied because the independent variable is in the
exponent.
x
xf 2
Let’s look at the graph of
this function by plotting
some points.
x 2x
3 8
2 4
1 2
0 1
-1 1/2
-2 1/4
-3 1/8
2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8
7
1
2
3
4
5
6
8
-2
-3
-4
-5
-6
-72
1
21 1
f
Recall what a
negative exponent
means:
BASE

13. x
xf 2
x
xf 3
Compare the graphs 2x, 3x , and 4x
Characteristics about the
Graph of an Exponential
Function where a > 1
x
axf
What is the
domain of an
exponential
function?
1. Domain is all real numbers
x
xf 4
What is the range
of an exponential
function?
2. Range is positive real numbers
What is the x
intercept of these
exponential
functions?
3. There are no x intercepts because
there is no x value that you can put
in the function to make it = 0
What is the y
intercept of these
exponential
functions?
4. The y intercept is always (0,1)
because a 0 = 1
5. The graph is always increasing
Are these
exponential
functions
increasing or
decreasing?
6. The x-axis (where y = 0) is a
horizontal asymptote for x -
Can you see the
horizontal
asymptote for
these functions?

14. The Rule of 72
 If a quantity is growing at rate r% per year (or month,
etc.) then the doubling time is approximately
(72 ÷ r) years (or months, etc.)
 For example, if a quantity grows at 8% per month, its
doubling time will be about 72 ÷ 8 = 9 months.

15. Ex: All of the properties of rational exponents apply
to real exponents as well. Lucky you!
Simplify:
3232
555
Recall the product of powers property,
am an = am+n

16. Ex: All of the properties of rational exponents apply
to real exponents as well. Lucky you!
Simplify:
10
2525
6
6)6(
Recall the power of a power property,
(am)n= amn

17. Application: Compound Interest
 Suppose:
- A: amount to be received
P: principal
r: annual interest (in decimal)
n: number of compounding periods per year
t: years
n
n
r
ptA 1)(
Example
 What would be the yield for the following investment?
P = 8000, r = 7%, n = 12, t = 6 years
612
12
07.0
18000A ≈ $12,160.84

18. References
 http://www.slideshare.net/itutor/exponential-
functions-24925841?qid=833b5856-9eca-411e-ab90-
c8e1a4352e6b&v=default&b=&from_search=3
 http://www.slideshare.net/jessicagarcia62/exponential
-functions-4772163?qid=833b5856-9eca-411e-ab90-
c8e1a4352e6b&v=default&b=&from_search=8
 http://www.slideshare.net/nclamelas/derivatives-of-
exponential-functions?qid=833b5856-9eca-411e-ab90-
c8e1a4352e6b&v=default&b=&from_search=15
 http://www.slideshare.net/dionesioable/module-2-
exponential-functions?qid=3a63ce8a-d910-43eb-b5b1-
8aac6231e021&v=qf1&b=&from_search=2
 http://www.slideshare.net/swartzje/ch-8-exponential-
equations-and-graphing?qid=3a63ce8a-d910-43eb-b5b1-
8aac6231e021&v=qf1&b=&from_search=7
Date accessed:03 March 2014