[MUSIC] Let's look at exponential graphs. For example, let's sketch the graph of

f(x) = 3 raised to the x minus 1st power minus 2.

And then we're going to find any x or y-intercepts of its graph.

Let's use graph transformations to help us here.

Let's start by sketching y = 3^x. So, if this is the y-axis and this is the

x-axis, what does y = 3^x look like? Well, it has

a y-intercept at 1 and then, when x is 1, y is 3^1 which is 3.

So, this is 2, here's 3. So, we have the point 1,3 lies on the

graph. And the exponential function looks like

this. Now, the x-axis or y = 0 is a horizontal

asymptote. Now, what does this -1 do here? What that

does is it shifts this graph rigidly 1 unit to the right.

That is the graph of y = 3^x-1 looks like this.

So, here's the y-axis, here's the x-axis. What's going to happen to this point over

here (0,1) if shift our graph one unit to the right?

This is going to move to 0 + (1,1) or (1,1).

And what's going to happen to this point here, 1,3? It's going to move to 1 +

(1,3) or (2 3). Now, let's plot these points.

This is x = 1 and y = 1. Here's our point 1,1 and if this is x =

2, y = 3, then here's our point (2,3).

And the horizontal asymptote will still remain at y = 0, so our graph will look

like this. Now, what does this -2 do to this graph?

What that does is it shifts this entire graph rigidly down two units.

So, let's say this is the y-axis and this is the x-axis.

What is going to happen to this point here, 1,1? It's going to move to 1 and

then 1 - 2 or 1, -1. And what's going to happen to this point

here? This 2,3? It's going to move to (2 3), - 2 or

(2,1). So, let's plot this points over here,

here is x = 1, y = -1.

So, here's our point (1,-1) and then here's 2 and 1.