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

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

-3-2-10123

.125.25.51248

.01563.0625.25141664

.00463

.02778.16667

1636216

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 ofy = (2x)+3

andy = 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) + kHey, 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 understandif we put some numbers in here.

Now we have a FormulaThe translation Th, k maps y = bx to y = b(x - h) + k

Let’s try a translation on ourbasic 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 gety = 2(x - 3) + 1

Step 1 y = 2(x - 3)

Step by = 2(x - 3) + 1

What happenedto the graph?

What happenedto 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 gety = 2(x + 4) - 2

Step 1 y = 2(x + 4)

Step b y = 2(x + 4) - 2

What happenedto the graph?

What happenedto 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 thegraph up k units

Negative k shifts thegraph down k unitsPositive h shifts thegraph left h units

Negative h shifts thegraph 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