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Problem 15 Medium Difficulty

Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3.

$ y = 1 - \frac{1}{2} e^{-x} $

Answer

We start with the graph of $y=e^{x}$ (Figure 16 ) and reflect about the $y$ -axis to get the graph of $y=e^{-x}$. Then we compress the graph vertically by a factor of 2 to obtain the graph of $y=\frac{1}{2} e^{-x}$ and then reflect about the $x$ -axis to get the graph of $y=-\frac{1}{2} e^{-x} .$ Finally, we shift the graph upward one unit to get the graph of $y=1-\frac{1}{2} e^{-x}$

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Video Transcript

okay, we're going to make a rough sketch emphasis on rough. And the point of that is to show that we understand the basic shape and that we understand the transformations that it's undergone without having to be too picky about all the details. So we're going to go through a series of transformations starting with y equals each of the X. But first I'm going to rearrange the terms in my function a little bit. I'm going to think of it as y equals negative 1/2 e to the negative X plus one. So I just switched the order of those two terms because it's going to help me to follow my transformations a little bit better. So I'm going to start with the basic function y equals eat of the X. You got your basic exponential Ruth. Then I noticed what happened is that we multiplied the X by negative one. So what that does to the graph is that reflects it across the Y axis. So now it looks like this looks more like exponential decay. And then I noticed what happened is that was multiplied by 1/2. So what that's going to do is make the graph half is tall. You're going to kind of make it look a little bit shallower. Then I noticed what happened is it was multiplied by a negative that's going to reflect it across the X axis. So now it looks like that now, all along here, each of these graphs have had a horizontal Assen tote at y equals zero. That's important to keep in mind for the next step, because the final transformation is that one was added. And when you add one, you shift the whole Graf up one, and that's going to shift the horizontal Assen tote up one as well. So graph is going to look like this. Now I'm going to make one minor adjustment to what I drew because I'm thinking about the Y intercept, and I don't really want to imply that the Y intercept here is 00 So if you think about it, what was the Y intercept As we went all along the way, he started out as one. It continued to be one, and then when we took half the height of the function, Now it's 1/2 and then when we flipped it upside down it was negative 1/2. And then when we shifted the graph up one, it would shift up to positive 1/2. So we're gonna take the, uh, and take that graph out of there. We're gonna shift it up just a little bit more so that it looks like that so we don't mislead and think that it goes through 00