Hey, everyone, and welcome back. Up to this point, we've spent a lot of time talking about functions. And in this video, we're going to be taking a look at a transformation of functions. This topic can seem a bit complicated at first because there are many different types of transformations that you'll see, but in this video, we're going to learn that transformations really only boil down to 3 basic transformations, which we're going to cover. After taking a look at these, I think you'll find that this concept is a lot less abstract and a lot clearer. So, let's get right into this. A transformation occurs when a function is manipulated such that it changes position or shape. Examples include the 3 main types of transformations that you're going to see: reflections, shifts, and stretches.

For a reflection, it occurs when a function is folded over a certain axis. If we were to take this function that we see right here and reflect it over the x-axis, it would look something like this; notice how we literally just took this graph and folded it—that's a reflection. Another type of transformation is a shift, which occurs when you move a function. If we were to take this function, which is currently at the position zero zero, the origin, and move it to some new location, the graph would look something like this. Notice how we literally took the function and moved it somewhere else—that's a shift. The last transformation we're going to look at is a stretch, which occurs when you imagine squeezing a function. So if you imagine taking this function and stretching it vertically such that it squeezes the function together, that's the idea of a stretch. So a stretch would look something similar to this from our original function.

These are the 3 basic transformations you're going to see throughout this course, and we will cover these in more detail as we go through this series on transformations, but it's also important to know how the function notation is going to change in these certain situations. So when you have a reflection, a reflection is going to become negative when you reflect over the x-axis. Notice how we started with f(x), and this became negative f(x). A shift is going to turn into this function, f(x-h)+k. And in this notation, the h represents the horizontal shift, and the k represents the vertical shift. For a stretch transformation, the function is going to look like this, where you have some constant multiplied by the function f(x). Here, c is the constant responsible for causing this kind of squeeze on the graph, or basically the vertical stretch.

Now let's see if we can actually apply this knowledge to an example. In this example, we're given the function f(x) = |x|, and we are also given the corresponding graph. What we're asked to do is match the following functions P of X, Q of X, and R of X to the correct corresponding graph because all of these functions that we have here are transformations of our original function, the absolute value of x. For this first function that I see, p(x) = |x-3| + 2, we need to figure out which one of these graphs this is associated with. This actually looks the most like a shift transformation because notice how we have this x-h+k, and here we have this x-3+2. I'm going to look for whichever one of these graphs appears to be a shift, and number 2 looks a lot like a shifted version of our original graph because we started here at the origin, and then we finished somewhere up here. So, I'm going to say that graph 2 matches with function a.

Now, let's take a look at function b. We have that q(x) = -|x|. Now, we first need to figure out what transformation from our original function this looks the most like. If I look at these transformations, this very much seems like a reflection, and a reflection is a situation where you're folding the graph over a certain axis, but we have a little bit of a problem here because both graphs 1 and 3 have been folded. We see reflections happening in both of these because our graph is originally pointed up, and we can see for both of these examples, the graph appears to be pointed down. But the difference is, for graph 3, it looks like the graph has also been squeezed, whereas for graph 1, it's just been flipped. So, looking at these graphs and looking at our function, I see that there's really no factor in front here that's going to cause it to be squeezed. So, that means that graph number 1 is going to match with option B. So that's our second function. Now I can tell just by process of elimination that for our 3rd function, function C, this is going to match with graph number 3, but I want us to understand why these two graphs match together as well. So, notice that we have this negative sign in our function, which is causing the fold or reflection over the x-axis, but I also notice this graph has gone through a vertical stretch, and that actually makes sense because we have a constant being multiplied by the front here as well. So because we have a constant and a negative sign, this is causing both a vertical stretch and a reflection transformation. It's very common that you're going to see multiple transformations happen to a single function. So that's just something you want to be aware of.

Overall, these are the 3 functions that match with the 3 graphs below. That is the basic idea of a transformation of a function, so hopefully, you found this helpful. Let me know if you have any questions, and thanks for watching.