Transforming exponential graphs | Mathematics III | High School Math | Khan Academy

We're told the graph of y = 2^x is shown below. All right, which of the following is the graph of y = 2^(-x) - 5?

So there's two changes here: instead of 2^x, we have 2^(-x) and then we're not leaving that alone; we then subtract five. So let's take them step by step.

So let's first think about what y = 2^x will look like. Well, any input we now put into it, x, we're now going to take the negative of it. So if I input a two, it's like taking the opposite of the two and then inputting that into 2^x.

And so what we're essentially going to do is flip this graph over the y-axis. So here we have the point (2, 4) over here; we're going to have the point (-2, 4). When x is zero, they're going to give us the same value, so they're both going to have the same y-intercept.

And so our graph is going to look like our graph is going to look something like this: they're going to be mirror images flipped around the y-axis. And so it's going to look like that. That is the graph of y = 2^x.

And then we have to worry about the subtracting five from it. Well, that's you're subtracting five from your final y-value. So that's going to—you're subtracting five to get your y-value now, or your y-value is going to be five lower, is I guess the best way to say it.

So this is going to shift the graph down by five. So instead of having the y-intercept there, it's going to be five lower. One, two... one each hash mark is two, so this is one, two, three, four, five—it's going to be right over there.

So shift down by five, two, four, five—it's going to look like that. And in the asymptote, instead of the asymptote going towards y = 0, the asymptote is going to be at y = 5. So the asymptote is going to be y = 5.

So it should look something like—it should look something like something like what I'm drawing right now—so something like that. So now we can look at which choices depict that.

So this first choice actually seems to be spot on. It's exactly what we drew. But we could look at the other ones just in case.

Well, this looks like—what did they do over here? It looks like instead of flipping over the y-axis, they flipped over the x-axis and then they shifted down, so that's not right. Here it looks like they got what we got, but then they flipped it over the x-axis.

And this looks like they flipped it over the y-axis, but then they shifted instead of shifting down by five; it looks like they shifted to the left by five. So we should feel pretty good, especially because we essentially drew this before even looking at the choices.