Pre-Calculus: Graphing Period, Amplitude, Shifts

The Trigonometric Functions
Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts
Fancy Graphing: Changes in Period, Amplitude, Vertical Shift, and Phase Shift Page [1 of 2]
Now let’s try some really fancy graphing and actually see that using these techniques, looking carefully at the period
and so forth, with a little bit of extra shifting, we can now graph really exotic functions. So, here’s an example:
How about if we want to graph y = -2 sine of x -
4
?
, and then all plus 1. It’s hard to even say it, let alone to look at it.
All right, well, what’s going on here? Well, as I see it, what we’ve got here is a regular sine function, but we’re going
to first of all, change the amplitude to 2. So, in fact, we going to kind of sort of stretch it, so it’s going to be a stretchy
kind of thing. The minus sign means I’m going to do a flippy kind of thing. Now what is the -
4
?
going to do? Well
that’s a shift in the x direction, in which way? Remember, add to x, go west, so I’m subtracting, so I’m going to go
east. So I’m going to go now to the right by
4
?
units and then when I’m done with all that stuff, I have to elevate the
entire graph one unit, because I’m adding to y. So, let’s just see roughly what this would like? This is real rough. So,
first of all, I take the sine and I look at x -
4
?
. So that’s now going to be a shift by
4
?
, so it’s sort of like that kind of
shift. Then I can’t do the minus two here because this is six, but I can do the minus sine and the minus sine just sort
of flips this thing over. So, now it looks sort of like this and then the plus one is going to elevate the whole thing one
unit and still look something like that. So, it’s going to look something weird. We’re going to move things around a lot.
Let’s try to do it actually now for real. Let’s set up some axes here and see how we make out. Now, the way I think
about this, by the way, is to draw in sort of all the intermediate functions lightly, so I’ll use sort of a light green
highlighter and then at the very end of the day after you’ve moved everything around a lot, then sort of put in the real
functions sort of in dark. So this has, by the way, a regular period of just 2? . So we’re going to do a full period here.
Now, I have an amplitude of 2, so we’re going to go, not to 1, but to 2. So, first I’m just going to graph -2 sine x, just to
get me going. So, -2 sine x, that’s going to be sort of negative of the sine function, standard period, but now I go all
the way down, to -2, so I don’t know if you’re going to be able to see this or not. I hope you sort of see it, but not too
clearly, because I’m going to moving this thing around. So this is just -2 sine of x.
Now, what do I do? What I want to look at is -2 sine of x -
4
?
? So what I want to do here now is shift a little bit to the
right by
4
?
. Now this is
2
?
, so that’s a little too much of a shift. I just want to shift sort of to here. So, this point will
now be here. This point down here will now be over here. This point will be here. This point will be here. And this
point will be up here. So, good luck. What a mess, huh? Sort of shifty business here. So this is just the next sort of
part of the graphing process. It keeps going down and then it just keeps going up like this now, and so on. So, now
it’s shifted over a little bit. And what’s the +1 do? It raises everything by one unit. So instead of being here at -2,
when I add 1, I’m going to be here at -1. This is zero, but now it would be at 1. This, at 2, is now going to be all the
way up to three, really high. This here at zero is now going to be at 1. So I’m just going to take that picture and move
it up. And now I’ll do it in nice dark font, so we can sort of see what’s going to happen here? So what’s going to
happen here is the following. It looks like this.
So, that is the graph. That is the final version of this graph. And notice, all I did was take a regular sine curve,
elongated it by 2, I flipped it by the negative, I shift it over by
4
?
and then I moved it up 1. Big mess, but there’s the
graph and it’s actually very accurate. But, I’m going to get rid of it really fast so you don’t look too closely.
Let’s try one more example, just for fun. How about let’s graph y = 2 x cosine ? x - 2? Now, here, notice we’re
changing a whole bunch of things. We’re going to change the amplitude and even though there is no shift, we are
changing the period and there is going to be a shift in the y direction.
The Trigonometric Functions
Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts
Fancy Graphing: Changes in Period, Amplitude, Vertical Shift, and Phase Shift Page [2 of 2]
So, let’s first of all figure out what that period is going to be. Well, let’s see that’s going to be 2? divided by the
absolute value of the coefficient, which is just ? , so this has a period of 2. That’s sort of peculiar because usually we
think of angles in terms of ? ’s and stuff because we’re measuring everything in terms of radians. But that’s just 2
radians. That’s okay. The units are now just going to be without the ? . That’s all right. Now, what do we have
here? We’re going to have a regular sine function, but we’re going to do a complete period within 2, and then we’re
going to expand the whole thing. We’re going to move the whole thing up by an amplitude of 2, and then we’ll shift.
So, let’s see if we can graph this one. In fact, let me put down some axes here in black first. Again, the trick is to put
in all the intermediate steps. I’m going to put in 2 here. Now, notice, I’m not putting in 2? . In fact, where would ?
be? Pi would be right around here, right? Because ? is 3.14 and I’m only going up to 2 here. So, this would be now
at 1, this would be at a half, this would be at three halves. There’s no ? ’s here because the period is 2. I’m going to
see a complete cycle here. Now I see, let me put in the 1, and a 2, and a -1, and a -2. So, I’m going to put in a
complete regular cosine that’s going to have a period 2, and amplitude 2 as well. So, we’re going to start way high
and go low. So, it starts way high, dives down, comes up, and dives up. That’s not what we want. That’s just the
graph of 2 cosine ? x. I’ve got to now subtract 2, which means I’ve got to lower this thing two units. So, in fact,
what’s that going to look like? Well, to show you that, I’m going to actually move this a little bit. Watch this. Here we
go. Let me just shift that up so you can really see what’s going on here. I’m going to take this now and I’m going to
subtract two, which means that this point, which used to be a 2, is now going to be at zero. This point, which used to
be at zero, is now going to be at -2. This point, which used to be at -2 is now going to go all the way down to -4. This
point here, which used to be a zero, is now going to be a -2 and this point, which used to be at 2 is now going to be
zero. So, I just take the whole picture and move it down and I get the final answer, which looks like this. It’s sort of
pretty because you’ll notice it has the feature that it just grazes the x-axis. That’s sort of cool, isn’t it? Because it used
to be a height of 2, which when I subtract 2, brings it down to a height of 1. So, in fact, this cosine curve is actually the
cosine curve that we wanted. This is y = 2 cosine ? x - 2. There’s the graph. Notice that, in fact, the period is indeed
2, as we hoped. The amplitude is 2, but I shifted everything down.
So you can see, even a really exotic graph like this can be graphed by just carefully picking off the pieces one by one
and then piecing it all together. It takes a little bit of practice to sort of get in the habit of doing these little sort of intergraphs,
before you get to the final product. But you can do it. And I want you to try it now. See you soon.