Hey, everyone, and welcome back. Up to this point, we've spent a lot of time talking about functions. In this video, we're going to be taking a look at a transformation of functions. Now, this topic can seem a bit complicated at first because there are a lot of 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 more clear.
So let's get right into this. A transformation occurs when a function is manipulated such that it changes position or shape. An example of this, and actually there are 3 main examples, are the 3 examples we have listed down here. The 3 basic types of transformations that you're going to see are reflections, shifts, and stretches. For a reflection, this occurs when a function is folded over a certain axis, so 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. So if we were to take this function, which is currently at the position 0, the origin, and we were to 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 transformation.
The last transformation we're going to look at is a stretch, and the stretch 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. A stretch would look something kind of like this from our original function. These are the 3 basic transformations you are 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, h represents the horizontal shift, and 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). So 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) is equal to the absolute value of x, and we are also given the corresponding graph. Now what we're asked to do is match the following functions P(x), Q(x), and R(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. So let's take a look at these. 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. And my question to you would be, what type of transformation do we have here? Because these are the 3 situations that we have transformations, and if I look at this, 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. So what I'm going to do is look for whichever one of these graphs appears to be a shift, and if I look at all of these, 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 P(x). Now let's take a look at function Q(x). We have that q(x) = -|x|.
Now we first need to figure out what transformation from our original function this looks the most like. And 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 notice 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 that for graph 3, it looks like the graph has also been squeezed, whereas for graph 1, it's just been flipped. And 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 Q(x).
So that's our second function. Now I can tell just by process of elimination that for our 3rd function, function R(x), 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. And 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. So overall, these are the 3 functions that match with the 3 graphs below. That is the basic idea of a transformation of function. So hopefully, you found this helpful. Let me know if you have any questions, and thanks for watching.
