Lesson Video: Function Transformations: Dilation Mathematics • Second Year of Secondary School

Video Transcript

In this video, we’ll learn how to identify function transformations involving horizontal and vertical stretches or compressions. After watching this video, you’ll be able to identify graphs of horizontal and vertical dilations or enlargements and how these transformations are described using function notation.

We’ll begin by recalling what we mean by a function. A function is like a machine. It has an input and an output. That output is achieved by performing an operation or a series of operations on the input. For example, take the function 𝑓 of 𝑥 equals two 𝑥 plus five. The input is 𝑥, and the output is two 𝑥 plus five. The operations performed by the machine are multiplying by two then adding five. Now, we want to look at how we achieve two types of transformations. These are horizontal dilations or enlargements — that is, enlargements parallel to the 𝑥-axis— and vertical dilations. Those are parallel to the 𝑦-axis.

Let’s take a function 𝑦 equals 𝑓 of 𝑥. With this function, we take a value of 𝑥, we substitute it into the function, and we get an output, a value for 𝑦 out. Imagine next that we multiply the entire function and the entire output by two. So, 𝑦 is equal to two times 𝑓 of 𝑥. What effect is multiplying the entire function by two have? Well, in this case, we substitute the value of 𝑥 in as normal. But we then take our output and we multiply that by two. This has the effect of doubling all of the outputs or all of the values of 𝑦. So what will that look like on the graph? Well, for each value of 𝑥, the output will be twice as large. So, if we take this point on the original curve, the value of 𝑦 will be twice as large. It will be up here somewhere.

The same can be said for this value on the original curve; the output will be twice the size. So it would be down here somewhere. And so, the graph will look a little something like this. Notice that the points where our graph crosses the 𝑥-axis remain unchanged. And that’s because at these points, the output 𝑦 is zero. So, if we double zero, we still get zero. The transformation that maps the function 𝑦 equals 𝑓 of 𝑥 onto the function 𝑦 equals two 𝑓 of 𝑥 is a vertical stretch. But we really call this a dilation by a scale factor of two parallel to the 𝑦-axis.

In general, we say that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑎 times 𝑓 of 𝑥 for real constants 𝑎 is a dilation by a scale factor of 𝑎 parallel to the 𝑦-axis. In other words, it’s a vertical stretch by a scale factor of 𝑎. But what about an alternative transformation? Let’s imagine now we have 𝑦 equals 𝑓 of two 𝑥. This time we multiply each value of 𝑥 by two and then substitute it into the formula. We’re substituting a value twice the size of the original into the function and then getting the relevant output. This feels a little bit like we’re speeding up the function and completing it twice as fast.

And so, this point on the original function will be completed in half the time. It will be here somewhere. Similarly, this point here will also be completed in half the time. It’s here somewhere. We can do the same with the remaining points, and we end up with this graph shown. This is a horizontal stretch or a dilation or enlargement by a scale factor, this time, of one-half parallel to the 𝑥-axis. We can say, in general, that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑏 times 𝑥 for real constants 𝑏 is a dilation by a scale factor one over 𝑏 parallel to the 𝑥-axis. In other words, it’s a horizontal compression. Now that we have these definitions, let’s look at some examples.

The function 𝑦 equals 𝑓 of 𝑥 is stretched in the vertical direction by a scale factor of one-half. Write, in terms of 𝑓 of 𝑥, the equation of the transformed function.

Another way of saying a stretch is to say that the function has been dilated or enlarged depending on which part of the world you’re in. And so, we recall that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑎 times 𝑓 of 𝑥 gives us that vertical stretch or dilation, and the scale factor is 𝑎. So, let’s compare this definition with our question. We have the function 𝑦 equals 𝑓 of 𝑥 being stretched in the vertical direction, and the scale factor is one-half.

So, comparing this to our definition, we see that we’re going to let 𝑎 be equal to one-half. Since 𝑦 equals 𝑎 times 𝑓 of 𝑥 gives us the vertical stretch by a scale factor of 𝑎, we get a vertical stretch by a scale factor of one-half by writing 𝑦 equals one-half of 𝑓 of 𝑥. Multiplying by one-half is the same as dividing by two. And so, we can alternatively say that, in terms of 𝑓 of 𝑥, the equation of our transformed function is 𝑦 equals 𝑓 of 𝑥 over two.

Let’s consider a similar example.

The function 𝑦 equals 𝑓 of 𝑥 is stretched in the horizontal direction by a scale factor of two. Write, in terms of 𝑓 of 𝑥, the equation of the transformed function.

Let’s recall how we achieve a stretch or a dilation of a function 𝑦 equals 𝑓 of 𝑥. We know that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑏 times 𝑥 gives us a horizontal stretch or dilation by a scale factor of one over 𝑏. So, let’s compare this definition to our question. We have a function 𝑦 equals 𝑓 of 𝑥 being stretched in the horizontal direction. The scale factor of this stretch or enlargement is two. Comparing this to our definition, we see that the scale factor is one over 𝑏, so we’re going to let one over 𝑏 be equal to two.

Let’s solve this equation for 𝑏 by first multiplying both sides by 𝑏. That gives us one equals two 𝑏. Then, we’ll divide both sides of this equation by two, giving us one-half equals 𝑏 or 𝑏 equals one-half. This means that for a horizontal dilation by a scale factor of two, our function becomes 𝑦 is equal to 𝑓 of a half 𝑥. Now, of course, multiplying by a half is the same as dividing by two, so we can write a half 𝑥 as 𝑥 over two. And this means the equation of our transformed function is 𝑦 equals 𝑓 of 𝑥 over two.

Let’s now look at how to identify a graph given the equation of the transformed function.

The figure shows the graph of 𝑦 equals 𝑓 of 𝑥. Which of the following is the graph of 𝑦 equals a half 𝑓 of 𝑥?

Let’s begin by looking at the equation of the transformed function. When we multiply by a scalar, that is, a real constant, that represents a dilation or enlargement of some description. In fact, when we multiply the entire function 𝑓 of 𝑥 by some scalar, we get a vertical dilation or enlargement by a scale factor of that number. And so, here we’re going to stretch the original graph vertically by a scale factor of one-half. That’s going to look like a vertical compression. To identify the correct graph, we’ll identify some of the key points on our graph.

Firstly, let’s consider this point here. It passes through the 𝑦-axis at two. When we compress our graph or stretch it vertically by a scale factor of one-half, this will now pass through a value of 𝑦 half the size. It’s going to pass through the 𝑦-axis at zero, one. Similarly, let’s take the point at 1.5, negative 0.6. We’re going to halve the value of the 𝑦-coordinate. The 𝑥-coordinate still remains unchanged, so it’s going to be 1.5, negative 0.3. And so, it’s going to look a little something like this. If we compare this to the graphs we’ve been given, we see that the only one that matches this criteria and the only one in fact that passes through the 𝑦-axis at one is (B). So, (B) is the graph of 𝑦 equals a half 𝑓 of 𝑥.

Let’s see if we can identify the equations of the other graphs. Looking at graph (A), we can see it’s actually been stretched by a scale factor of two. And so, the equation of this one must be 𝑦 equals two times 𝑓 of 𝑥. Graph (C), however, has been compressed by a scale factor of a half. But this time, that’s in the horizontal direction. To achieve a horizontal dilation by a scale factor of one-half, we need to multiply the values of 𝑥 by two. So, the equation of this graph is 𝑦 equals 𝑓 of two 𝑥.

Then, if we look at graph (D), we see something similar has occurred. This time it’s stretched in a horizontal direction but by a scale factor of two. To achieve this, we need to multiply all the values of 𝑥 by one-half. So, graph (D) is 𝑦 equals 𝑓 of a half 𝑥. And graph (E) is a different beast altogether. This represents a combination of stretches. It stretched vertically by a scale factor of two and horizontally by a scale factor of two. And so, its equation is, in fact, a combination of (A) and (D). It’s 𝑦 equals two times 𝑓 of a half 𝑥. The correct answer here, though, is (B).

We’ll now have a look at another example that involves recognizing dilations of functions graphically.

The red graph in the figure represents the equation 𝑦 equals 𝑓 of 𝑥, and the green graph represents the equation 𝑦 equals 𝑔 of 𝑥. Express 𝑔 of 𝑥 as a transformation of 𝑓 of 𝑥.

Let’s begin by just comparing the red graph and the green graph in our diagram. Both graphs pass through the origin, the point zero, zero. So, let’s compare this point, this sort of relative maximum, on the red graph to the relative maximum on the green graph. The value of the 𝑦-coordinate remains unchanged. The value of the 𝑥-coordinate, however, is doubled. It’s transformed from roughly negative 1.25 to negative 2.5. Let’s consider this local minimum and compare it to the local minimum on the green graph. Once again, the value of the 𝑦-coordinate remains unchanged. The 𝑥-part of the coordinate, though, is doubled. It’s gone from 0.5 on the red graph to one, roughly, on the green graph.

So, what transformation maps the red curve onto the green curve? Well, we can see it’s been stretched parallel to the 𝑥-axis. We call that a horizontal dilation or enlargement. Since everything in the 𝑥-direction has been doubled, we can say this horizontal dilation is by a scale factor of two. So, how do we represent this using function notation? Well, we recall that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑏 times 𝑥 for real constants 𝑏 represents a horizontal stretch by a scale factor of one over 𝑏. Let’s compare our scale factor with the scale factor in the general definition.

When we do, we see that we’re going to let one over 𝑏 be equal to two. To find the value of 𝑏 then, let’s multiply both sides of this equation by 𝑏 and then divide through by two so that one-half is equal to 𝑏 or 𝑏 is equal to one-half. We can therefore say that the green graph is 𝑦 equals 𝑓 of a half 𝑥. But, of course, we said that the green graph is given by the equation 𝑦 equals 𝑔 of 𝑥. So we can say that 𝑔 of 𝑥 must be equal to 𝑓 of a half 𝑥. We can simplify a little by writing a half 𝑥 as 𝑥 over two. And so, we see that 𝑔 of 𝑥 as a transformation of 𝑓 of 𝑥 is 𝑔 of 𝑥 equals 𝑓 of 𝑥 over two.

In our next example, we’ll look at how to find coordinates of a point on a graph following a dilation or enlargement.

The figure shows the graph of 𝑦 equals 𝑓 of 𝑥 and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 equals 𝑓 of two 𝑥.

Let’s begin by recalling the transformation that maps the graph of 𝑦 equals 𝑓 of 𝑥 onto the graph of 𝑦 equals 𝑓 of two 𝑥. We know that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑏 times 𝑥 is a horizontal dilation by a scale factor of one over 𝑏. And so, if we compare our equation, that’s 𝑦 equals 𝑓 of two 𝑥, to the general equation, 𝑦 equals 𝑓 of 𝑏𝑥, we can see that we’re going to have a horizontal dilation. Let’s work the scale factor out by letting 𝑏 be equal to two.

When we do, we see that for our function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of two 𝑥 is a horizontal dilation by a scale factor of one-half. In other words, we’re going to compress our graph about the 𝑦-axis. When we do, it looks a little something like this. We’ll call the coordinate of our local maximum on our transformation 𝐴 prime as shown. We can see that its 𝑦-coordinate remains unchanged, but its 𝑥-coordinate is halved. So, 𝐴 prime is two over two, one or one, one. And so, we see that the corresponding local maximum for the transformation 𝑦 equals 𝑓 of two 𝑥 is one, one.

In our final example, we’ll look at how these processes can help us to draw a full transformation by dilation.

The diagram shows the graph of the function 𝑦 equals 𝑓 of 𝑥 for values of 𝑥 greater than or equal to negative three and less than or equal to three. Sketch the graph of 𝑦 equals one-third of 𝑓 of 𝑥 on the same set of axes.

Let’s begin by recalling what transformation maps 𝑦 equals 𝑓 of 𝑥 onto 𝑦 equals a third 𝑓 of 𝑥. We know that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑎 times 𝑓 of 𝑥 gives us a vertical dilation by a scale factor of 𝑎. Comparing this with the function in our question, we see we’re going to have a vertical dilation, but we’re going to let 𝑎 be equal to one-third. And so, for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 is a third 𝑓 of 𝑥 is a vertical dilation by a scale factor of one-third. Let’s do this point by point.

We’re going to start with the point on the left-hand side of our diagram. This point has coordinates negative three, three. By performing a vertical dilation or a stretch, we find a third of the 𝑦-coordinate. A third of three is one, so the corresponding coordinate on our transformed graph is negative three, one. Since it’s being stretched vertically, it will still pass through the 𝑥-axis in the same place. And what about the point with coordinates zero, negative three. Once again, the 𝑥-coordinate remains unchanged, and we divide the 𝑦-coordinate by three to get zero, negative one.

We can perform the same process for the point one, negative three, and that will map onto the point with coordinates one, negative one. It will pass through the 𝑥-axis in the same place again. And finally, we’ll transform the point with coordinates three, one. We divide the 𝑦-coordinate by three, and we find this point has coordinates three, a third. And so, we see that the graph of 𝑦 equals a third 𝑓 of 𝑥 is as shown. It’s a vertical dilation or a stretch by a scale factor of one-third.

In this video, we’ve learned that we can perform dilations or enlargements in one of two ways. For a function 𝑦 equals 𝑓 of 𝑥, 𝑦 is 𝑎 times 𝑓 of 𝑥 for real constants 𝑎 gives us a vertical dilation by a scale factor of 𝑎. Similarly, 𝑦 is equal to 𝑓 of 𝑏𝑥 represents a horizontal dilation or a stretch parallel to the 𝑥-axis. And this time, the scale factor is one over 𝑏.