Lesson Video: Transformation of Trigonometric Functions Mathematics

In this video, we will learn how to translate or stretch the trigonometric function and find the rule of a trigonometric function given the transformation.

17:53

Video Transcript

In this lesson, we will learn how to translate or stretch the trigonometric function and find the rule of the trigonometric function given the transformation. So what you’re going to be able to do after this lesson is find the coordinates of a point in a trigonometric graph after it’s been transformed. We’re also gonna be able to translate the graph of the trigonometric function in directions of the 𝑥- and 𝑦-axis. And we’re also gonna stretch the graph of a trigonometric function.

So the first thing we need to remember is actually what our trigonometric graphs look like. And we can see that on this first page, what we’ve got is the graph of 𝑦 equals tan 𝑥, 𝑦 equals sin 𝑥, and 𝑦 equals cos 𝑥. So we need to remember these when we’re going along and looking at the problems in this lesson.

Now, before we get on and have a look at some questions, what we need to do is remind ourselves of how the transformations of graphs work. So we’re gonna remind ourselves how we transform graphs. And if we remember, what we’re looking at in this lesson are translations and stretches. So if we start with translations, then what we have is 𝑓 of 𝑥 plus 𝑎 or 𝑓 of 𝑥 plus 𝑎. But how do these differ?

Well, first of all, if we have a look at what 𝑓 of 𝑥 plus 𝑎 means, well, what this is is in fact a shift of the graph in the 𝑥-axis of negative 𝑎. So what this means in practice is that if we took the 𝑥-coordinates of our first translation, then we’d subtract 𝑎 from them to get the translation. For our second translation, what we’d do is look at our 𝑦-coordinates and we’d add 𝑎 to them.

Now, there’s another little tip for remembering our translations or in fact any transformation. If the transformation is taking place inside our parentheses, then it’s gonna involve the 𝑥-axis. And also we do the opposite of what we might think. So we can see in this case, instead of adding 𝑎 as it says, we would subtract 𝑎. However, if it’s outside the parentheses, then it’s involving the 𝑦-axis. And we do what we’d expect. So here we add 𝑎 like we said. We’d add 𝑎 to our 𝑦-coordinates.

Now, when we’re looking at trig graphs, the shift in the 𝑥-axis is known as a phase shift and the shift in the 𝑦-axis is known as a vertical shift. Well, here’s a quick example. So if we’ve got the graph here in pink of 𝑦 equals cos of 𝑥 or the cosine of 𝑥, well then if we shift it 90 degrees to the right, so 90 degrees in the 𝑥-axis, then what we’re gonna have is the graph 𝑦 equals cos. Then we’ve got 𝑥 minus 90 cause like we said it’s gonna be within the parentheses because we’re dealing with the 𝑥-axis. And because we’ve added 90, then what we’re gonna do is subtract 90. And that gives us our translation.

So in fact, what we’ve done is we’ve had a phase shift of 90 degrees in the 𝑥-axis. But what we can also see is that this phase shift actually is meant that our graph of 𝑦 equals cos 𝑥 has become the graph 𝑦 equals sin 𝑥.

Okay, great, so now let’s move on to our stretches. So with stretches, what we have is that 𝑓 of 𝑎𝑥 is a stretch parallel to the 𝑥-axis with a scale factor of one over 𝑎 and 𝑎𝑓 of 𝑥 is a stretch parallel to the 𝑦-axis with a scale factor of 𝑎. So once again, we can see that when we’re looking at the 𝑥-axis, it does the opposite of what you might think. So rather than multiplying by the number, we actually divide because the scale factor is one over 𝑎, whereas with the 𝑦-axis, we multiply by that 𝑎.

So in practice, what this means is that if we’re looking at the first stretch, then we’d divide each of our 𝑥-coordinates by 𝑎. And what this is is actually a change in the period of our trigonometric graph. And the period of a trigonometric graph is the distance between our peaks or from any point to the next matching point, whereas if we look at the second stretch, then what we do in practice is multiply the 𝑦-coordinates by 𝑎. And what this does is this in fact changes the amplitude of our graph. And what the amplitude is is the height from the center line to the peak or to the trough. Or we can measure the height from the highest to the lowest points and divide that by two.

So now let’s take a quick look at an example of a stretch. So here we’ve got 𝑦 equals sin 𝑥. So then, what we’ve drawn on is 𝑦 equals sin two 𝑥. And we can see here that each of the 𝑥-coordinates are in fact halved. And therefore, in turn, we can see that our period is halved because the period of our first function, which was 𝑦 equals sin 𝑥, is 360 degrees. And we can see that we go from one trough to the other trough is 360 degrees or from one corresponding point to the other corresponding point 360 degrees, whereas if we take a look at the blue line, so 𝑦 equals sin two 𝑥, then the distance between our peaks is 180 degrees.

Okay, great, so we now reminded ourselves what the transformation of graphs are for our trigonometric graphs. Now, let’s get on and solve some problems.

The figure shows the graph of 𝑓 of 𝑥. A transformation maps 𝑓 of 𝑥 to 𝑓 of 𝑥 minus three. Determine the coordinate of 𝐴 following this transformation.

So if we’ve got a transformation that maps 𝑓 of 𝑥 to 𝑓 of 𝑥 minus three, then the transformation we’re taking a look at is in fact going to be a shift or a translation. And if we quickly remind ourselves of our translations, then we know that 𝑓 of 𝑥 plus 𝑎 is a shift in the 𝑦-axis of 𝑎, whereas 𝑓 of 𝑥 plus 𝑎 is a shift in the 𝑥-axis of negative 𝑎.

So therefore, what we’re looking at is our first translation or our first shift. And what this is going to be is going to be a shift in the 𝑦-axis of 𝑎. So it’s called a vertical shift. Well, 𝑎 in our scenario is going to be negative three. So what this means in practice is that we subtract three from the 𝑦-coordinates of our point. So in fact, what we can see is that our point 𝐴 is gonna shift three units down in the 𝑦-axis. So therefore, we can say that the coordinates of 𝐴 following the transformation are going to be 45, negative two.

Okay, great, so now let’s take a look at another example. Now, in this next example, what we’re going to do is actually use the transformation we looked at earlier in the introduction. However, we’re gonna focus on one particular point during this transformation.

The figure shows the graph of 𝑓 of 𝑥. A transformation maps 𝑓 of 𝑥 to 𝑓 of two 𝑥. Determine the coordinates of 𝐴 following this transformation.

Well, what we’re looking at in this problem is a transformation that maps 𝑓 of 𝑥 to 𝑓 of two 𝑥. So in fact, what we’re looking at is a stretch. Well, if we remind ourselves of our stretches, then we know that 𝑎𝑓 of 𝑥 is a stretch parallel to the 𝑦-axis with a scale factor of 𝑎 and 𝑓 of 𝑎𝑥 is a stretch parallel to the 𝑥-axis with a scale factor of one over 𝑎. So we can see that, in fact, we’re looking at the second scenario because we’re looking at a stretch parallel to the 𝑥-axis because we’ve got 𝑓 of two 𝑥. Well, because we’ve got 𝑓 of two 𝑥, we can see that the scale factor is going to in fact be a half. So what this means in practice is that each of our 𝑥-coordinates is going to be halved.

So what we’ve done here is actually sketched how this would look on the graph. So therefore, we can say that the coordinates of 𝐴 following the transformation are going to be 90, negative one. We can see this as the corresponding point on our graph. Or we could’ve found this by just, as we said, halving the 𝑥-coordinate, so halving 180, which gives us our 90, 90, negative one.

Okay, so great, now what we’re gonna do is move on to an example where we have a selection of graphs and we have to choose which one is the correct one for a transformation of one of our trigonometric functions.

Which of the following is the graph of 𝑦 equals cos 𝑥 minus 90?

Well, first of all, let’s remind ourselves what the graph of 𝑦 equals cos 𝑥 looks like. And we can see it shown here. So this is the shape of our graph. Now, what we can see is that what we’re looking for is 𝑦 equals cos 𝑥 minus 90. And what this is is gonna be a translation or a shift of the 𝑦 equals cos 𝑥 graph. And that’s cause we can think of it as the transformation 𝑓 of 𝑥 plus 𝑎, where this type of translation is a shift in the 𝑥-axis of negative 𝑎 units. Well, in fact, because our 𝑎 is negative 90, then negative 𝑎 is gonna give us positive 90. So what we’re gonna do is shift our graph 90 degrees to the right, so it’s 90 degrees in our 𝑥-axis, which we’ve shown here in our sketch.

Now, all what we need to do is identify which one of our graphs is the same as this. Well, in fact, we can see that graph (A) matches this. And this is the correct graph because this is showing the phasal shift that we had of 90 degrees. And we can see that the other graphs are incorrect because graph (B) shows the graph in fact of 𝑦 equals cos 𝑥, graph (C) is a period change, graph (D) is in fact a period change and a phase shift, and graph (E) is a vertical shift. So we can confirm that the correct graph is graph (A).

Okay, so now what we’re gonna take a look at is another example of shift. But this time, what we’re gonna look at is a problem where we’ve got vertical shift or translation in the 𝑦-axis.

Which of the following is a graph of 𝑦 equals cos 𝑥 plus one?

So what we can see here is in fact 𝑦 equals cos 𝑥 plus one is gonna be a translation of 𝑦 equals cos 𝑥. And we know this is a translation because it’s in the form 𝑓 of 𝑥 plus 𝑎, where this is a shift in the 𝑦-axis of 𝑎. So we now we’re gonna have 𝑎 units shifted in the 𝑦-axis. But what’s it a translation of?

Well, what it is going to be is a translation of 𝑦 equals cos 𝑥. And we know what it’s gonna be is a shift in the 𝑦-axis of one unit. But what does this mean in practice? Well, what it means is we actually add one to each of our 𝑦-coordinates. Well, to help us work out which one of our graphs is going to be this shift of one unit in the 𝑦-axis, what we’ve done here is sketched 𝑦 equals cos 𝑥. And we’ve done it onto the graph (A).

Well, actually if we shift it one unit in the 𝑦-axis, so we add one to each of the 𝑦-coordinates, we can see that it’d map itself onto the graph that is shown in (A). Because instead of the peaks being at one, they would be at two. And instead of the troughs being at negative one, they would be at zero. So therefore, we can say that the graph (A) is the graph of 𝑦 equals cos 𝑥 plus one. Well, if we wanted to check the other graphs, we can see that these are incorrect because (B) is in fact the graph 𝑦 equals cos 𝑥. We’ve got (C), which is a phase shift; (D) is a period change; and (E) is a period change and a phase shift.

So now what we’ve had a look at is some questions which identify which graph matches a particular equation. But for the next one, what we need to do is actually decide which equation is correct for the graph we’re looking at.

The figure shows the graph of a function. Which of the following equations represents the graph? (A) 𝑦 equals sin two 𝑥, (B) 𝑦 equals sin 𝑥 plus two, (C) 𝑦 equals two sin 𝑥, (D) 𝑦 equals sin 𝑥 minus two, or (E) 𝑦 equals sin 𝑥 plus two.

So to answer this problem, what we’ve done is drawn a quick sketch of a part of our sine graph. That’s the sort of first part that we have that’s positive. And what we can see is that the peak of 𝑦 equals sin 𝑥 is at one. And if we carried it on a little bit, so we actually took it back from zero to the negative side, we can see that our trough or one of our troughs would be at negative one. However, if we take a look at the graph that we’ve got here, then we can see that the peak is at negative one. And in fact, the trough is at negative three. So we can see that there’s been a vertical shift downwards two units.

So what we can do is remind ourselves about our translations with graphs. So we know that 𝑓 of 𝑥 plus 𝑎 is a vertical shift of 𝑎 units or a shift in the 𝑦-direction. Then we’ve got 𝑓 of 𝑥 plus 𝑎 is a phase shift or a shift in the 𝑥-direction of negative 𝑎 units. So therefore, if we’re just considering the shift, we could say that it would be 𝑦 equals sin 𝑥 minus two because we’re looking at our first translation. But what we could think is, well, hold on, is there going to be a stretch as well?

Well, in fact, we don’t know what the 𝑥-coordinates are actually in because they’re not in degrees. So it’s difficult to tell whether a stretch has occurred or not. Well, if we take a look at (C), we can see for (C) we’ve got a stretch here. And it’s gonna be a stretch in the 𝑦-direction. So therefore, we know that this cannot be the correct answer because actually the amplitudes of both of our graphs are exactly the same, cause the amplitudes are both one.

Well, if we take a look at our other stretch, we can see that this is a stretch parallel to the 𝑥-axis. And here there isn’t a shift applied as well. So we’ve just got the stretch. We’ve already identified that a shift has taken place. So this cannot be the correct answer. So therefore, we can say that (D) is the equation which represents our graph. And if we have a look at (B) and (E), well, we can see that (B) would be incorrect because that’d be a shift in the 𝑥-direction. So it’d be a phase shift. But we said that ours is a vertical shift. And (E) would be incorrect cause for (E) what we would’ve done is actually moved the graph two units up instead of two units down. So this would’ve been incorrect as well.

Okay, we looked at a number of different examples, and they’ve all looked at a single transformation. For the final question, we’re actually gonna look at a double transformation and see what this does.

Which of the following is the graph of 𝑦 equals sin 𝑥 over four minus one?

So in this problem, what we’re looking at is a combination of transformations. However, because one of our transformations is horizontal and one of our transformations is vertical, it doesn’t actually matter which order we carry out our transformations in. Well, if we look at our first part of our transformation, this is a stretch because it’s in the form 𝑓 of 𝑎𝑥. This is where the stretch is parallel to the 𝑥-axis with a scale factor of one over 𝑎, whereas we can see that the second part of our transformation is in fact a translation. And that’s because it’s in a form 𝑓 of 𝑥 plus 𝑎. So it’s a vertical shift 𝑎 units.

So therefore, what we’ve got, like I said, was something that actually refers to a horizontal direction because it’s parallel to the 𝑥-axis. And we’ve got something else which is a shift, and that’s a vertical shift. So that’s actually looking at our 𝑦-axis. Well, as we’ve said, this graph is gonna be a transformation of the graph 𝑦 equals sin 𝑥. So what we’ve done here is sketched on the graph (C), our graph of 𝑦 equals sin 𝑥. So we can see this is what it looks like.

Well, straight away, we can see that it goes through the origin: zero, zero. So therefore, as we can see from our transformation, we’re gonna apply a vertical shift of negative one because we’re gonna subtract one from each of the 𝑦-coordinates. So therefore, the point that goes through the origin in our transformed graph is gonna go to zero, negative one. Well, therefore, what we can do is rule out graphs (A) and (E). And that’s because we can see that neither of these graphs go through the point zero, negative one.

Okay, but we’ve still got three graphs in the running. So what do we need to do now? However, we can also rule out graphs (C) and (D). And that’s because even though they go through the point zero, negative one, we can see that in fact a phase shift would’ve have to have taken place to actually transform them from 𝑦 equals sin 𝑥 because they don’t go through it at the same point on the graph. Because for both of those graphs, the trough is at zero, negative one. So we can definitely say graph (B) is the correct graph.

But what we can also do to confirm it’s the correct graph is have a look at distance between the point where it crosses the 𝑦-axis and its first peak. And on the original graph of 𝑦 equals sin 𝑥, this is 90. Well, if we look at the new graph, it’s 360, which makes sense because the scale factor should be one over 𝑎 for our stretch. Well, one over one over four is the same as multiplying by four. 90 by four is 360.

So we’ve now had a look at a number of examples. So now let’s take a look at the key points of the lesson. So first of all, in the lesson, we took a look at translation. So if we’ve got 𝑓 of 𝑥 plus 𝑎, this is a shift parallel to the 𝑥-axis of negative 𝑎 units, so a phase shift. And 𝑓 of 𝑥 plus 𝑎 is a shift parallel to the 𝑦-axis of 𝑎 units, known as a vertical shift. And then we took a look at stretches. And we know that if we’ve got 𝑓 of 𝑎𝑥, this is a stretch in the 𝑥-direction with a scale factor of one over 𝑎. And if we have 𝑎𝑓 of 𝑥, this is a stretch in the 𝑦-direction with a scale factor of 𝑎. And the first one is a period change, and the second one causes an amplitude change.

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