[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.