Pre-Calculus: Graph Sine, Cosine with Coefficients

This lesson has not been reviewed.

Please purchase the lesson to review.

The Trigonometric Functions
Graphing Sine and Cosine Functions
Graphing Sine or Cosine Functions with Different Coefficients Page [1 of 3]
Let’s see how we get to look at variations on the theme. Remember, that the sine function just sort of has this very
nice, basic shape, very wiggly, and the cosine function has the similar shape but starts of high and comes down low.
So this is what sort of the take home lesson is so far, that sine just looks like this, goes between 1 and -1, and has a
period of 2? and the cosine function has a similar shape, but we start up high and come down low. Again, period of
2? and it wiggles like this. What happens if we modify these functions a little bit? Well, if we modify the functions, of
course, the picture is going to change. But how?
Well, let’s start off by taking something that looks like this: y = 2 x sin(x). Well, let’s just try to sketch this. We take
every single value that we used to get and we multiply it by two to get the current value. So, let me put here, 2? . I’m
just going to sketch in a real light rough sketch of just the good old fashion -- our friend -- the sine curve. So first I’m
just going to graph that. If I graph that, what do we see? And it keeps going. That’s not the answer; that’s just the
sine of x graphed. Now I’ve got to take each value and multiply it by 2. So, for example, 0 x 2 is still 0. So these
points are still going to be the zeroes. But this point, instead of being a 1 now, it’s going to be way high up here at 2.
And this is going to be all the way down here at -2. So what I see is just that it sort of stretches it a little bit. This is
the graph of y = 2sin(x). So, when you have a coefficient in front like that, that actually changes the amplitude. And
the amplitude is nothing more than the absolute value of that coefficient. So the amplitude is the absolute value of 2,
which in this case is 2. In general, if I write y = a sin(x), the amplitude of that will be the absolute value of a. And it
tells you where the highest point is going to be and where the lowest points are going to be. So, it changes the
amplitude. In fact, if you listen to radio, if you’re really cheesy, you might listen to --or talk radio -- you might listen to
AM. That stands for amplitude modulation. That means that when the signal waves, the radio waves go through,
those are actually sine waves and in fact, the difference in the different waves is amplitude. Changes in amplitude
actually produce AM radio signals. Okay, so that’s fine, that’s sort of no problem.
Let’s graph another one just to practice this. Suppose I asked you to graph y = -
1 cos
2
x . Well, the first thing I always
do is just graph the standard function sort of, as is. And then try to just distort it in the appropriate way. So, here’s
let’s say, 2? and I cut that in half and then in half again. Here’s 1, here’s -1. So, I’m going to first just graph in the
cosine function. Here’s a little sketch of the cosine function. It keeps going. So, there’s the cosine function. Okay,
now what am I doing here? I’m putting a -
1
2
. Now, you know what a negative sign in front of this thing does. It flips
it. So, in fact, I’m going to flip this like this, but what’s the
1
2
going to do? It’s going to take the 1 and bring it down to
an amplitude of
1
2
. So, in fact, now I’m going to shrink, but I’m also going to flip. So, it’s a flip and a shrink. So if I do
it all in one step, this point, which used to be 1, now is at -
1
2
. So, in fact, what I see here is the following. The zeroes
are still the same, but now it’s a reflection because the minus sign and the
1
2
means that my height now is at
1
2
.
And this low point is at -
1
2
. So, with a small number in there, it shrinks the amplitude. A larger number will make the
amplitude longer, bigger. A negative sign just slips over the x axis, which we saw before in graphing.
Okay, now what if we change the period? Well, now let’s suppose we look at y = sin(2X). So, if we look at sin(2),
what would that be? Well, the amplitude is still 1, so I still go between 1 and -1, but what’s the effect of that 2 in the x?
Well, let’s think about that. Let me graph the good old fashioned sine function first. So, here is going to be 2? ,
here’s ? , here’s
2
?
, 3
2
?
, and so forth. So, let’s just graph the good old-fashioned sine function. So I go up to 1 and
down to -1, I keep going in a very nice way. Okay, but now what’s going to happen? Well, what’s going to happen is
The Trigonometric Functions
Graphing Sine and Cosine Functions
Graphing Sine or Cosine Functions with Different Coefficients Page [2 of 3]
all the x values get doubled. So, for example, here,
2
?
, when I plug that in, I’m going to be looking at the sine of ? .
So, I’m going to sort of already be over here. And when I plug in ? , I’m going to get sin(2? ) so I’m already going to
be here. So, basically, by the time I get to ? , I’ve already done a complete cycle, right? Because, at
2
?
, I’m sort of
at ? . By the time I get to ? , I’m already at 2? , so I’ve taken this whole thing and sort of squashed it into this period
here. So, in fact, the period here should be just ? , right? Because everything has been -- all my values are doubling,
which means that I should see the whole picture in half as much space since everything is doubled. And so, in fact,
this tells me how to find the period. If I want to find the period, which means the length that I go to see one complete
cycle, that actually is going to always be the following: It’s going to be 2? ÷ whatever that coefficient is. So, if in
general, I have sin(b x x ), then I put the absolute value of b down there. And that tells me the period. So, in this
case, I put a 2 there, I see 2
2
?
, that’s ? . That means just within ? , I should see a complete period. It should go up
and come down, so I’m taking this whole picture and shrinking it in. So, if I do that, what I now have to do is
compactify everything and you see how I did a whole period just in ? , so it keeps going now, of course, it always
keeps going. And you can see that within this length, in fact, I have two periods, because I multiply everything by 2.
So, in fact, now I’m changing the period or the frequency of how often we see these waves, and in fact this is FM,
frequency modulation. The amplitude doesn’t change at all, but it’s the amount of waves we have in one particular
period that changes. So, in general, you can always the period by looking at this simple thing, taking 2? divided by
the absolute value of the coefficient.
Let’s try an example here. Another one. Let’s take a look at y = cos
1
4
x. Oh, by the way, it’s a little shorthand here.
Sometimes, in fact a lot of times, people just write cosine of one-fourth x. And then it’s clear that it’s the cosine of that
whole thing. It’s not cosine of
1
4
times x. It’s cosine of all that stuff there. Well, what would the period be here?
Well, the recipe always is take 2? and divide it by whatever that number is in absolute values. In this case, it’s
1
4
.
When you invert and multiply, you see 8? . That means I’m going to do one complete period in 8? . So, in fact,
instead of 2? , which I usually see, I’m going to see now 8? . It’s going to be four times the time just to get one entire
period. So a small number here means the period is going to be sort of drawn out; it will be a big period, right? I’ve
got to wait a long time to get a complete cycle. A large number here means I’m going to get those cycles very, very
early. So, if I graph this, I’m graphing cosine. Now let’s see. Let’s put 2? right here. In fact, that might not even be
enough. This is going to be 4? , this will be 6? , I just barely make it. This is 8? . Let me just put in really lightly, the
good old fashioned cosine function, just to help us out. So that means one period right here, so it looks sort of like this
and it keeps going -- I won’t put them all in. But there’s one complete period of cosine. But now this person actually
has a complete period all the way stretched out to 8? , which means that this point here sort of corresponds to the
halfway point here and so this point corresponds to that point, sort of half of the half, and so on. So, I’ve got to stretch
out the whole picture so it’ll look like this, really stretchy, same basic shape, but now stretched out over a length of
8? because the period is 8? . So, really stretched, not compared to the first one which will just sort of wiggle a lot,
as you can see. So you can find a period by just taking 2? and dividing it by the absolute value of that.
Let’s do one last one together and put these ideas together. Y = -2cos
1
2
x. Let’s graph that together. Okay, so what
would I do? Well, the first thing that I would do is figure out what the period is. What’s the period? Well, to find the
period, I would take 2? and I’d divide it by the coefficient, so that’s 2? ÷
1
2
= 4? . So that means I have to wait
4? in order to see 1 complete cycle. Okay? What’s the amplitude? The amplitude is 2 so I don’t just go to -- let’s
say this is 2? , this would be ? , this would be 3? , and this would be 4? -- so instead of just having our usual
The Trigonometric Functions
Graphing Sine and Cosine Functions
Graphing Sine or Cosine Functions with Different Coefficients Page [3 of 3]
cosine thing right here, I’m going to have to go all the way out to here to get my complete cycle. The 2 means, in fact,
that I’m going to be at an amplitude of 2, so not 1. So, I’m going to go very high and very low, and that negative sign
tells me I’m going to flip the picture. So, now you have to put all these pieces together, so where would the zeroes
be? Well, the zeroes are going to be here, half of half, and then half of the second half. Here it’s going to be all the
way, let’s see, reverse of what you think, so all the way up to 2. We start off at -2, and we end at -2, and so we get a
picture that looks like this. So, it’s a really large, and it keeps going, of course. It’s a really large cosine curve. It has
period 4? , you have to wait 4? to get a complete cycle. And the amplitude is 2, so it starts way down at negative 2
and goes way the heck up to 2 and then so forth. And then you can see where the zeroes are. They’re going to be
half of half, and half of this half. So the zeroes are at ? , 3? , and all the odd multiples of ? .
Anyway, that’s putting a whole bunch of ideas together. You have a sense of now, the period, finding the period, and
the amplitude and you can graph these little basic things. Up next, we will take a look at more graphics of more exotic
things. I’ll see you there.