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# Graphing exponential functions

CCSS.Math:

## Video transcript

- [Voiceover] We're told, use the interactive graph below to sketch a graph of y is equal to negative two, times three to the x, plus five. And so this is clearly an exponential function right over here. Let's think about the behavior as x is, when x is very negative, or when x is very positive. When x is very negative, three to a very negative number like let's say you had three to the negative 3rd power, that would be 1/27th, or three to the negative 4th power, that would be 1/81st. So this is going to get smaller, and smaller, and smaller. It's going to approach zero as x becomes more negative. And since this is approaching zero, this whole thing right over here is going to approach zero. And so this whole expression is, if this first part is approaching zero, then this whole expression is going to approach five. So we're going to have a horizontal asymptote that we're going to approach as we go to the left. As x gets more and more negative, we're going to approach positive five. And then as x gets larger, and larger, and larger, three to the x is growing exponentially. But then we're multiplying it times negative two, so it's going to become more, and more, and more negative. And then we add a five. And so what we have here, well this looks like a line. We want to graph an exponential. So let's go pick the exponential in terms of x. There you have it. And so we can move three things. We can move this point, it doesn't even need to just be the y-intercept, although that's a convenient thing to figure it out. We can move this point here. And we can move the asymptote. And maybe the asymptote's the first interesting thing. We said as x becomes more, and more, and more, and more negative, y is going to approach five. So let me put this up here. So that's our asymptote. It doesn't look like it quite yet, but when we try out some values for x and the corresponding y's, and we move these points accordingly, hopefully our exponential is going to look right. So let's think about, let's pick some convenient x's. So let's think about when x is equal to zero. If x is equal to zero, three to the zeroth power is one. Negative two times one is negative two, plus three is three. So when x is equal to zero, y is three. And let's think about when x is equal to one, and I'm just picking that 'cause it's easy to compute. Three to the first power is three, times negative two is negative six, plus five is negative one. So when x is one, y is negative one. And so let's see, does this, is this consistent with what we just described? When x is very negative, we should be approaching, we should be approaching positive five, and that looks like the case. As we move to the left, we're getting closer, and closer, and closer to five, in fact, it looks like they overlap, but it's really, we're just getting closer, and closer, and closer 'cause this term, this term right over here is getting smaller, and smaller, and smaller as x becomes more, and more, and more negative. But then as x becomes more and more positive, this term becomes really negative 'cause we're multiplying it times a negative two, and we see that it becomes really negative, so I feel pretty good about what we've just graphed. We've graphed the horizontal asymptote, it makes sense, and we've picked two points that sit on this, on the graph of this exponential. So I can check my answer, and we got it right.