 The lesson is on transformation of functions. You'll already have in your notes all the different types of transformations that you can do on graphs. And we will pick out certain ones to do in this lesson. The first one we will do is y is equal to sine x plus 2. This is of the type f of x plus a. That means a is a vertical shift up 2. So to graph this, we will do nothing more than get what is our normal horizontal axis and move it up 2. And we will graph a sine wave because we have not changed that part at all as we did before. And two periods go from negative 2 pi to 2 pi. And so on our horizontal axis, that's where we have all our zero points. So we'll do the same here. And one is on zero. The other one is on pi. The other one is on 2 pi. And then we have one at negative pi and one at negative 2 pi. And from negative 2 pi, we will go up 1 and then down 1 in between the points and then up 1 and down 1. And this will be our graph. Most of the time you just want a nice sketch of your graph just to know exactly where it lies and how it looks. Now if we look at the domain and range of this, the domain on the sine function has not changed. It's negative infinity to infinity. But the range has changed. It has gone from what was before normally for a sine wave, negative 1 to 1, 2, 1 to 3. So in that it has moved up, our new range becomes 1 to 3. Now let's go to another type. Here's the example. f of x is equal to 1 over 2x. Let's rewrite that to 1 half times 1 over x. So this one actually is a vertical shrink of 1 half and is of the type 1 over a f of x. Now to graph that, we are going to graph our reciprocal function and we are going to shrink it vertically by a half. So to do that, we will still have the asymptotes along the y-axis and along the x-axis. And we will have our points a little differently than before. Okay, the point that was usually 1, 1 now becomes 1, 1 half. The point that was usually 2, 1 half now becomes 2 and 1 quarter. And then at 1 half, our point will be at 1. And you will see it's the same reciprocal function, almost looks the same as before 2. That's only half of it. We have to do the other side. Our point that now is at negative 1, negative 1 half, at negative 2, negative 1 quarter, and at negative 1 half, negative 1. For this one, the domain is still x cannot be equal to 0, that vertical asymptote. And for the range, y cannot equal 0. And that's the horizontal asymptote here. So let's go on to another one. Example 3, we have f of x is equal to square root of 3x. This is one that is not common and some people get very confused by it. But this is of the type f of a of x. It's a horizontal shrink by one-third. And that's what we have to remember. It's a shrink horizontally by one-third. So if we were to graph this one, we normally have the point 0, 0 on the original. Of course, that's the same here, 0, 0. But instead of having 1, 1, we have to do one-third 1 as a point. There's our shrink. And another point we can graph on this is 3, 3. So with those points, we can actually graph this. Our domain on this is still the same as the original 1, which is 0 to infinity. And our range is still the same. Nothing has changed on that, 0 to infinity. So those are three basic transformations. Now let's try a combination of different transformations. Our example is 2 to the negative x plus 1 plus a 3. This is a combination of a reflection across the y-axis, a shift. And in this case, we have to change things around to look at what that shift is. So let's move this to 2 to the negative x minus 1 and then plus the 3. So it's a shift, right, 1, and then a shift vertically up 3. So to graph that, we will have to do our exponential function, move it up 3. And again, we can just change that axis to move it up 3. So along the horizontal axis, that will be our asymptote. We will have a point at 1, 4, 0, 5, and negative 1, 7. And putting all that together, we will have a graph that looks like this. Now the domain on our exponential function has not changed, so it's negative infinity to infinity. Our range, of course, has changed because we have moved up 3. So that becomes 3 to infinity. And that concludes our lesson on transformations.