1. Exponential Functions
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
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.

2. Graphs of Exponential Functions
Let’s look at the graph of y = 2x
That was easy

3. Comparing Graphs of Exponential Functions
What happens to the graph of y = bx as the value of b changes?
Now, let’s look at the graphs.
Let’s look at some tables of values.
I think I see a pattern here.

4. Let’s Look at the Other Side
What happens to the graph of y = bx when b < 1 and the value of b changes?
Now, let’s look at the graphs.
Let’s look at some tables of values.
I knew there was going to be a pattern!

5. 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)

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

7. Let’s take a closer look
If y = bx and y = f(x), then f(x) = bx
This is a little confusing, but I’m sure it gets easier.
If the translation Th, k maps y = f(x) to y = f(x - h) + k
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.

8. 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.

9. 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)
What happened
to the graph?
Step b
y = 2(x - 3) + 1
What happened
to the graph now?
What conclusions can we make from this example?

10. 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)
What happened
to the graph?
Step b
y = 2(x + 4) - 2
What happened
to the graph now?
What conclusions can we make from this example?

11. 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
This translation stuff sounds pretty shifty, but don’t let it scare you.
Negative k shifts the
graph down k units
Positive h shifts the
graph left h units
Negative h shifts the
graph right h units

12. 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