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 youโre going to be able to
do after this lesson is find the coordinates of a point in a trigonometric graph
after itโs been transformed. Weโre also gonna be able to
translate the graph of the trigonometric function in directions of the ๐ฅ- and
๐ฆ-axis. And weโre 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 weโve got is the graph of ๐ฆ equals tan ๐ฅ, ๐ฆ equals sin ๐ฅ, and ๐ฆ
equals cos ๐ฅ. So we need to remember these when
weโre 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 weโre gonna remind ourselves how
we transform graphs. And if we remember, what weโre
looking at in this lesson are translations and stretches. So if we start with translations,
then what we have is ๐ of ๐ฅ plus ๐ or ๐ of ๐ฅ plus ๐. But how do these differ?
Well, first of all, if we have a
look at what ๐ of ๐ฅ plus ๐ means, well, what this is is in fact a shift of the
graph in the ๐ฅ-axis of negative ๐. So what this means in practice is
that if we took the ๐ฅ-coordinates of our first translation, then weโd subtract ๐
from them to get the translation. For our second translation, what
weโd do is look at our ๐ฆ-coordinates and weโd add ๐ to them.
Now, thereโs another little tip for
remembering our translations or in fact any transformation. If the transformation is taking
place inside our parentheses, then itโs gonna involve the ๐ฅ-axis. And also we do the opposite of what
we might think. So we can see in this case, instead
of adding ๐ as it says, we would subtract ๐. However, if itโs outside the
parentheses, then itโs involving the ๐ฆ-axis. And we do what weโd expect. So here we add ๐ like we said. Weโd add ๐ to our
๐ฆ-coordinates.
Now, when weโre looking at trig
graphs, the shift in the ๐ฅ-axis is known as a phase shift and the shift in the
๐ฆ-axis is known as a vertical shift. Well, hereโs a quick example. So if weโve got the graph here in
pink of ๐ฆ equals cos of ๐ฅ or the cosine of ๐ฅ, well then if we shift it 90 degrees
to the right, so 90 degrees in the ๐ฅ-axis, then what weโre gonna have is the graph
๐ฆ equals cos. Then weโve got ๐ฅ minus 90 cause
like we said itโs gonna be within the parentheses because weโre dealing with the
๐ฅ-axis. And because weโve added 90, then
what weโre gonna do is subtract 90. And that gives us our
translation.
So in fact, what weโve done is
weโve had a phase shift of 90 degrees in the ๐ฅ-axis. But what we can also see is that
this phase shift actually is meant that our graph of ๐ฆ equals cos ๐ฅ has become the
graph ๐ฆ equals sin ๐ฅ.
Okay, great, so now letโs move on
to our stretches. So with stretches, what we have is
that ๐ of ๐๐ฅ is a stretch parallel to the ๐ฅ-axis with a scale factor of one over
๐ and ๐๐ of ๐ฅ is a stretch parallel to the ๐ฆ-axis with a scale factor of
๐. So once again, we can see that when
weโre looking at the ๐ฅ-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 ๐, whereas with the
๐ฆ-axis, we multiply by that ๐.
So in practice, what this means is
that if weโre looking at the first stretch, then weโd divide each of our
๐ฅ-coordinates by ๐. 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 ๐ฆ-coordinates by ๐. 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 letโs take a quick look at
an example of a stretch. So here weโve got ๐ฆ equals sin
๐ฅ. So then, what weโve drawn on is ๐ฆ
equals sin two ๐ฅ. And we can see here that each of
the ๐ฅ-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 ๐ฆ
equals sin ๐ฅ, 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 ๐ฆ equals sin two ๐ฅ, 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, letโs get on and solve some
problems.
The figure shows the graph of ๐ of
๐ฅ. A transformation maps ๐ of ๐ฅ to
๐ of ๐ฅ minus three. Determine the coordinate of ๐ด
following this transformation.
So if weโve got a transformation
that maps ๐ of ๐ฅ to ๐ of ๐ฅ minus three, then the transformation weโre 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 ๐ of ๐ฅ plus ๐ is a shift in the ๐ฆ-axis of
๐, whereas ๐ of ๐ฅ plus ๐ is a shift in the ๐ฅ-axis of negative ๐.
So therefore, what weโre 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 ๐ฆ-axis of ๐. So itโs called a vertical
shift. Well, ๐ in our scenario is going
to be negative three. So what this means in practice is
that we subtract three from the ๐ฆ-coordinates of our point. So in fact, what we can see is that
our point ๐ด is gonna shift three units down in the ๐ฆ-axis. So therefore, we can say that the
coordinates of ๐ด following the transformation are going to be 45, negative two.
Okay, great, so now letโs take a
look at another example. Now, in this next example, what
weโre going to do is actually use the transformation we looked at earlier in the
introduction. However, weโre gonna focus on one
particular point during this transformation.
The figure shows the graph of ๐ of
๐ฅ. A transformation maps ๐ of ๐ฅ to
๐ of two ๐ฅ. Determine the coordinates of ๐ด
following this transformation.
Well, what weโre looking at in this
problem is a transformation that maps ๐ of ๐ฅ to ๐ of two ๐ฅ. So in fact, what weโre looking at
is a stretch. Well, if we remind ourselves of our
stretches, then we know that ๐๐ of ๐ฅ is a stretch parallel to the ๐ฆ-axis with a
scale factor of ๐ and ๐ of ๐๐ฅ is a stretch parallel to the ๐ฅ-axis with a scale
factor of one over ๐. So we can see that, in fact, weโre
looking at the second scenario because weโre looking at a stretch parallel to the
๐ฅ-axis because weโve got ๐ of two ๐ฅ. Well, because weโve got ๐ of two
๐ฅ, 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 ๐ฅ-coordinates is going to be halved.
So what weโve done here is actually
sketched how this would look on the graph. So therefore, we can say that the
coordinates of ๐ด following the transformation are going to be 90, negative one. We can see this as the
corresponding point on our graph. Or we couldโve found this by just,
as we said, halving the ๐ฅ-coordinate, so halving 180, which gives us our 90, 90,
negative one.
Okay, so great, now what weโre
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 ๐ฆ equals cos ๐ฅ minus 90?
Well, first of all, letโs remind
ourselves what the graph of ๐ฆ equals cos ๐ฅ 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
weโre looking for is ๐ฆ equals cos ๐ฅ minus 90. And what this is is gonna be a
translation or a shift of the ๐ฆ equals cos ๐ฅ graph. And thatโs cause we can think of it
as the transformation ๐ of ๐ฅ plus ๐, where this type of translation is a shift in
the ๐ฅ-axis of negative ๐ units. Well, in fact, because our ๐ is
negative 90, then negative ๐ is gonna give us positive 90. So what weโre gonna do is shift our
graph 90 degrees to the right, so itโs 90 degrees in our ๐ฅ-axis, which weโve 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 ๐ฆ equals cos ๐ฅ,
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 weโre gonna take
a look at is another example of shift. But this time, what weโre gonna
look at is a problem where weโve got vertical shift or translation in the
๐ฆ-axis.
Which of the following is a graph
of ๐ฆ equals cos ๐ฅ plus one?
So what we can see here is in fact
๐ฆ equals cos ๐ฅ plus one is gonna be a translation of ๐ฆ equals cos ๐ฅ. And we know this is a translation
because itโs in the form ๐ of ๐ฅ plus ๐, where this is a shift in the ๐ฆ-axis of
๐. So we now weโre gonna have ๐ units
shifted in the ๐ฆ-axis. But whatโs it a translation of?
Well, what it is going to be is a
translation of ๐ฆ equals cos ๐ฅ. And we know what itโs gonna be is a
shift in the ๐ฆ-axis of one unit. But what does this mean in
practice? Well, what it means is we actually
add one to each of our ๐ฆ-coordinates. Well, to help us work out which one
of our graphs is going to be this shift of one unit in the ๐ฆ-axis, what weโve done
here is sketched ๐ฆ equals cos ๐ฅ. And weโve done it onto the graph
(A).
Well, actually if we shift it one
unit in the ๐ฆ-axis, so we add one to each of the ๐ฆ-coordinates, we can see that
itโd 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 ๐ฆ equals cos ๐ฅ 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
๐ฆ equals cos ๐ฅ. Weโve 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 weโve 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 weโre looking
at.
The figure shows the graph of a
function. Which of the following equations
represents the graph? (A) ๐ฆ equals sin two ๐ฅ, (B) ๐ฆ
equals sin ๐ฅ plus two, (C) ๐ฆ equals two sin ๐ฅ, (D) ๐ฆ equals sin ๐ฅ minus two, or
(E) ๐ฆ equals sin ๐ฅ plus two.
So to answer this problem, what
weโve done is drawn a quick sketch of a part of our sine graph. Thatโs the sort of first part that
we have thatโs positive. And what we can see is that the
peak of ๐ฆ equals sin ๐ฅ 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 weโve 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 thereโs been a
vertical shift downwards two units.
So what we can do is remind
ourselves about our translations with graphs. So we know that ๐ of ๐ฅ plus ๐ is
a vertical shift of ๐ units or a shift in the ๐ฆ-direction. Then weโve got ๐ of ๐ฅ plus ๐ is
a phase shift or a shift in the ๐ฅ-direction of negative ๐ units. So therefore, if weโre just
considering the shift, we could say that it would be ๐ฆ equals sin ๐ฅ minus two
because weโre 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 donโt know what
the ๐ฅ-coordinates are actually in because theyโre not in degrees. So itโs difficult to tell whether a
stretch has occurred or not. Well, if we take a look at (C), we
can see for (C) weโve got a stretch here. And itโs gonna be a stretch in the
๐ฆ-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 ๐ฅ-axis. And here there isnโt a shift
applied as well. So weโve just got the stretch. Weโve 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 thatโd be a shift in the
๐ฅ-direction. So itโd be a phase shift. But we said that ours is a vertical
shift. And (E) would be incorrect cause
for (E) what we wouldโve done is actually moved the graph two units up instead of
two units down. So this wouldโve been incorrect as
well.
Okay, we looked at a number of
different examples, and theyโve all looked at a single transformation. For the final question, weโre
actually gonna look at a double transformation and see what this does.
Which of the following is the graph
of ๐ฆ equals sin ๐ฅ over four minus one?
So in this problem, what weโre
looking at is a combination of transformations. However, because one of our
transformations is horizontal and one of our transformations is vertical, it doesnโt
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 itโs in the form ๐ of ๐๐ฅ. This is where the stretch is
parallel to the ๐ฅ-axis with a scale factor of one over ๐, whereas we can see that
the second part of our transformation is in fact a translation. And thatโs because itโs in a form
๐ of ๐ฅ plus ๐. So itโs a vertical shift ๐
units.
So therefore, what weโve got, like
I said, was something that actually refers to a horizontal direction because itโs
parallel to the ๐ฅ-axis. And weโve got something else which
is a shift, and thatโs a vertical shift. So thatโs actually looking at our
๐ฆ-axis. Well, as weโve said, this graph is
gonna be a transformation of the graph ๐ฆ equals sin ๐ฅ. So what weโve done here is sketched
on the graph (C), our graph of ๐ฆ equals sin ๐ฅ. 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, weโre gonna apply a vertical shift of negative one because weโre
gonna subtract one from each of the ๐ฆ-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 thatโs because we can see that
neither of these graphs go through the point zero, negative one.
Okay, but weโve 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 thatโs because even though they
go through the point zero, negative one, we can see that in fact a phase shift
wouldโve have to have taken place to actually transform them from ๐ฆ equals sin ๐ฅ
because they donโt 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
itโs the correct graph is have a look at distance between the point where it crosses
the ๐ฆ-axis and its first peak. And on the original graph of ๐ฆ
equals sin ๐ฅ, this is 90. Well, if we look at the new graph,
itโs 360, which makes sense because the scale factor should be one over ๐ for our
stretch. Well, one over one over four is the
same as multiplying by four. 90 by four is 360.
So weโve now had a look at a number
of examples. So now letโs 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 weโve got ๐ of ๐ฅ plus ๐,
this is a shift parallel to the ๐ฅ-axis of negative ๐ units, so a phase shift. And ๐ of ๐ฅ plus ๐ is a shift
parallel to the ๐ฆ-axis of ๐ units, known as a vertical shift. And then we took a look at
stretches. And we know that if weโve got ๐ of
๐๐ฅ, this is a stretch in the ๐ฅ-direction with a scale factor of one over ๐. And if we have ๐๐ of ๐ฅ, this is
a stretch in the ๐ฆ-direction with a scale factor of ๐. And the first one is a period
change, and the second one causes an amplitude change.