Exponential Functions
• The domain is the set of real numbers, and the range is the
set of positive real numbers
• if b > 1, the graph of y = bx rises from left to right and
intersects the y-axis at (0, 1). As x decreases, the negative
x-axis is a horizontal asymptote of the graph.
• If 0 < b < 1, the graph of y = bx falls from left to right and
intersects the y-axis at (0, 1). As x increases, the positive
x-axis is a horizontal asymptote of the graph.
The equation y = bx
is an exponential function provided that
b is a positive number other than 1. Exponential functions
have variables as exponents.
Generalizations about Exponential Functions
Adham hegazy

Graphs of Exponential Functions
Let’s look at the graph of y = 2x
x 2x
-3
-2
-1
0
1
2
3
.125
.25
.5
1
2
4
8
Let’s look at the graph of y = (½)x
x (½)x
-3
-2
-1
0
1
2
3
8
4
2
1
.5
.25
.125
That
was
easy

Comparing Graphs of
Exponential Functions
What happens to the graph of y = bx
as the value of b changes?
x 2x
4x
6x
-3
-2
-1
0
1
2
3
.125
.25
.5
1
2
4
8
.01563
.0625
.25
1
4
16
64
.00463
.02778
.16667
1
6
36
216
Let’s look at
some tables
of values.
Now, let’s look
at the graphs.
I think I see a
pattern here.

Let’s Look at the Other Side
What happens to the graph of y = bx
when b < 1 and the value of b changes?
x (½)x
(¼)x
-3
-2
-1
0
1
2
3
8
4
2
1
.5
.25
.125
64
16
4
1
.25
.0625
.01563
Let’s look at
some tables
of values.
Now, let’s look
at the graphs.
I knew there
was going to
be a pattern!

Let’s Shift Things Around
Let’s take another look at the graph of y = 2x
Now, let’s compare
this to the graphs of
y = (2x
)+3
and
y = 2(x+3)
x
-3
-2
-1
0
1
2
3
(2x
)+3
3.125
3.25
3.5
4
5
7
11
2(x+3)
1
2
4
8
16
32
64

Translations of Exponential Functions
The translation Th, k maps y = f(x) to y = f(x - h) + k
Hey, that rings a
bell!
Since y and f(x) are the Sam Ting,
It looks like that
evaluating functions
stuff.
Remember if f(x) = x2
,
then f(a - 3) = (a - 3)2
It’s all starting
to come back
to me now.
We can apply this concept to the
equation y = bx

Let’s take a closer look
If the translation Th, k maps y = f(x) to y = f(x - h) + k
If y = bx
and y = f(x), then f(x) = bx
This is a little confusing,
but I’m sure it gets easier.
Then the translation Th, k maps y = bx
to y = b(x - h)
+ k
This will be easier to understand
if we put some numbers in here.

Now we have a Formula
The translation Th, k maps y = bx
to y = b(x - h)
+ k
Let’s try a translation on our
basic exponential equation
Let’s apply the transformation
T3, 1 to the equation y = 2x
The transformed equation would be y = 2(x - 3)
+ 1
I’m not ready to push the easy button yet.
Let’s look at some other examples first.

Let’s look at some graphs
Let’s start with the graph of y = 2x
Let’s go
one step
at a time.
When the
transformation
T3, 1 is applied
to the equation
y = 2x
we get
y = 2(x - 3)
+ 1
Step 1 y = 2(x - 3)
Step b y = 2(x - 3)
+ 1
What happened
to the graph?
What happened
to the graph now?
What conclusions can we
make from this example?

Let’s look at some other graphs
Let’s start with the graph of y = 2x
Let’s go
one step
at a time.
When the
transformation
T-4, -2 is applied
to the equation
y = 2x
we get
y = 2(x + 4)
- 2
Step 1 y = 2(x + 4)
Step b y = 2(x + 4)
- 2
What happened
to the graph?
What happened
to the graph now?
What conclusions can we
make from this example?

Let’s Summarize Translations
The translation Th, k maps y = bx
to y = b(x - h)
+ k
Positive k shifts the
graph up k units
Negative k shifts the
graph down k units
Positive h shifts the
graph left h units
Negative h shifts the
graph right h units
This translation
stuff sounds
pretty shifty,
but don’t let it
scare you.

This exponential equation
stuff is pretty easy.
I feel
like
jumping
for joy!
Oh my! I think
I’ll just push the
easy button.
That
was
easy