Starting with the graph of $ y = e^x $, find the equation of the graph that results from
(a) reflecting about the line $ y = 4 $ .
(b) reflecting about the line $ x = 2 $.
all right, This is an interesting problem because we're being asked to do reflections that we don't typically do. Typically, we would reflect across the X axis or the y axis, but not a line like Y equals four or X equals two. So I think it will be very helpful if we start by sketching the situation. So we have the graph of y equals E to the X, which typically looks like this, and it has a horizontal Assen tote at a height of zero, and it has a Y intercept of one. Now, if we're going to reflect that across the line, X equals four and this might not all be drawn to scale here, but just to give us an idea, the line y equals four. Here's the line Y equals four. So what we're going to see is that we have ah, horizontal Assen tote four units away from that at a height of eight. And we're going to see that we have a Y intercept at a height one unit from that seven. So the reflected graph will look like this. All right, now, our goal is to figure out the equation of that red one. So what could we have done to the green one, the original to get the red one? Well, we obviously have flipped it upside down so we could have gone from why Equals each of the X two y equals the opposite of each of the X. That would be a reflection across the X axis. And that would take us to to here with the Y intercept of negative one. But then we have to ship that whole thing up eight units so that it will have a y intercept of seven. So if we shift that whole thing up, eight units we're going to have why equals the opposite of each of the X Plus eight. Now, after doing something that like that, that's kind of unusual and atypical. I would recommend going head and grabbing your graphing calculator and typing that in and taking a look at the graph and making sure that that does look like we took y equals e to the X and reflected it about the line Y equals four. Now for part B, Let's see what it's going to look like if we reflected about the line X equals two so again we'll start with our standard y equals each of the X, which has a Y intercept of one, and we'll go ahead and we'll find the line. X equals two. So if we reflected across that line, we would have something like this and we would have a point. That's the reflection of the 0.0.1 That point would be 41 Okay, so what could we do to our original graph to get to that? Well, it looks like we have to turn it around horizontally, so we need to reflect it across the Y axis. So that's going to give us the equation. Why equals e to the negative X or you to the opposite of X if we reflect our original graph across the Y axis. So now we have this, and then we're going to have to move that thing four units to the right so that instead of having a Y intercept or a point at a height of one, we have a point at a height of 14 units to the right. So that means we're going to go ahead and add actually subtract four from X just from X leave the negative on the outside. That's the confusing part. And if you're not sure about that, you can try either putting a plus four or minus four and then graphing it in your calculator and making sure that you get it to look like it with the correct direction.