Video Transcript
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.
