About this video
Sal graphs y = 2^(-x) - 5 from the graph of y = 2^x. Replacing x with -x flips the graph over the y-axis, and subtracting 5 shifts it down 5, moving the horizontal asymptote to y = -5. He then rules out answer choices that flip the wrong way or shift the wrong direction.
You'll be able to
- explain why replacing x with -x reflects an exponential graph over the y-axis.
- shift an exponential graph down and find its new horizontal asymptote.
- match a transformed exponential equation to its graph.
Key words
- exponential function
- A function like y = 2^x where the variable is in the exponent.
- horizontal asymptote
- A horizontal line the graph gets closer and closer to but never reaches.
- y-intercept
- The point where a graph crosses the y-axis, found by setting x = 0.
Transcript
- [Voiceover] We're told the graph of y equals two to the x is shown below, alright. Which of the following is the graph of y equals two to the negative x minus five? So there's two changes here. Instead of two to the x, we have two to the negative x and then, we're not leaving that alone, we, then, subtract five.
So let's take 'em step by step. So let's first think about what y equals two to the negative x would look like. Well, any input we now put into an x, we're now going to take the negative of. So if I input a two, it's like taking the opposite of the two and then, inputting that into two to the x.
And so, what we're essentially going to do is flip this graph over the y-axis. So, here we have the point two comma four. Over here, we're going to have the point negative two comma four. 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 are 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. So, it's going to look like that. That is the graph of y is equal to two to the negative 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 work. You're subtracting five to get your y-value now, where 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, is going to be right over there. So shift down by five, two, four, five. It's going to look like that. And the asymptote, instead of the asymptote, going towards y equals zero, the asymptote is going to be at y is equal to negative five.
So the asymptote is going to be y equals negative five. So it should look something like, something like what I am drawing right now. Something like, something like that. So now we can look at which choices. So this should be the graph of y equals two to the negative x minus five.
So let's see which of these choices depict that. So this first choice actually seems to be spot-on. It's exactly what we drew. Well, we can look at the other ones, just in case. Well, this is, this looks like, what did they do over here? Looks like they, instead of flipping over the y-axis, they took the, 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 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 'cause we essentially drew this before even looking at the choice.
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