 In this video, we provide the solution to question number 23 for the practice final exam for math 1050. We have to graph the function f of x equals negative e to the x minus one plus two. And we're gonna graph this using transformations. We need to list the transformations. So our basic graph here, what's the basic graph? It's the function y equals e to the x, which of course is gonna look something like this, right, this exponential function. Okay, that's the basic graph. What are the transformations in play here? Well, what does this negative sign do? That negative sign means we're going to reflect, we're gonna reflect across the x-axis, okay? This x minus one suggests that we're gonna shift the graph right by a factor of one. And this plus two means we actually are gonna shift the graph up by two. Now with exponential functions like I drew it on the board a moment ago, one of the things to pay attention with the exponential is that if there's no transformations in play here, an exponential function will have a y intercept at y equals one. And it'll have this upward direction assuming the base is greater than one in that situation. E, you know, it's about 2.7, that's the case. There is a horizontal asymptote, of course, at the x-axis. This is when it's untransformed. So based upon the transformations, these things might change. If we reflect across the x-axis, this y intercept will then come down to negative one, the vertical asymptote is left alone. If we shift things to the right, right, that moves that point over there, and if we shift things up, that moves the point, it also moves the asymptote. So let's pay attention to those things. I'm gonna first draw the asymptote for an exponential. So like I said, if your asymptote is the x-axis, a reflection across the x-axis doesn't do anything. If you're the x-axis and you shift everything to the right by one, it doesn't move, it doesn't affect a horizontal line, so no big deal. But going up by two does affect the horizontal asymptote here. So because of the shift, our horizontal asymptote is now going to be at y equals two. When graphing an x-adventure function, you definitely should list its horizontal asymptote. If any of these graphs have asymptotes, they should be on the graph illustrated here. So let's pay attention to that, where that happened to that y-intercept again, right? So it started off at y equals one, we then reflected down, it becomes y equals negative one. We shifted to the right by one, and then we shifted up by two. So in the end, we ended up at the point one comma one. So that's the, we have to at least list one point, so that's the point we listed right there. Then we have to draw the rest of it. It should still have the same basic exponential shape. It's been shifted. It hasn't been stretched at all, but it was reflected. So instead of going upwards, it should go downwards to match up with this point we have right here. On the left-hand side, it should still be going off towards its horizontal asymptote, trying to draw that effectively like so. And then otherwise it'll go down like so. And so this is the graph then of F, where the three transformations are listed here to the right.