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.
