This document discusses exponential functions and their key characteristics: 1. Exponential functions have the independent variable in the exponent. 2. They have domains of all real numbers and ranges of positive real numbers. 3. They do not have x-intercepts but always have a y-intercept of (0,1). 4. Their graphs are always increasing and have a horizontal asymptote at y=0 as x approaches negative infinity.

2. 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
-7
( )
2
1
21 1
==− −
f
Recall what a
negative exponent
means:
BASE

3. ( ) 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?

4. All of the transformations that you
learned apply to all functions, so what
would the graph of
look like?
x
y 2=
32 += x
y
up 3
x
y 21−=
up 1
Reflected over
x axis 12 2
−= −x
y
down 1right 2

5. x
y −
= 2
Reflected about y-axis This equation could be rewritten in
a different form: x
x
x
y 





=== −
2
1
2
1
2
So if the base of our exponential
function is between 0 and 1
(which will be a fraction), the
graph will be decreasing. It will
have the same domain, range,
intercepts, and asymptote.
There are many occurrences in nature that can be
modeled with an exponential function. To model these
we need to learn about a special base.

6. The Base “e” (also called the natural base)
To model things in nature, we’ll
need a base that turns out to be
between 2 and 3. Your calculator
knows this base. Ask your
calculator to find e1
. You do this by
using the ex
button (generally you’ll
need to hit the 2nd or yellow button
first to get it depending on the
calculator). After hitting the ex,
you
then enter the exponent you want
(in this case 1) and push = or enter.
If you have a scientific calculator
that doesn’t graph you may have to
enter the 1 before hitting the ex
.
You should get 2.718281828
Example
for TI-83

7. ( ) x
xf 2=
( ) x
xf 3=
( ) x
exf =

8. 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 43
=−x 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.
343 =−x We did not cancel the 2’s, We just used
the property and equated the exponents.
You could solve this for x now.

9. Let’s try one more:
8
1
4 =x The left hand side is 4
to the something but
the right hand side
can’t be written as 4 to
the something (using
integer exponents)
We could however re-write
both the left and right hand
sides as 2 to the something.
( ) 32
22 −
=
x
32
22 −
=x
So now that each side is written
with the same base we know the
exponents must be equal.
32 −=x
2
3
−=x
Check:
8
1
4 2
3
=






−
8
1
4
1
2
3
=
( ) 8
1
4
1
2 3
=

10. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au