>> Julie Harland:
This is from Math Gal
Julie Harland.
Please visit my website
at YourMathGal.com,
where all of my videos are
organized by topic.
This is Part 2
of Graphing Trig Functions.
In Part 1, we did a sketch
of y = sine of x
by making this table
of values, which we got
from the circular functions,
the unit circle.
And this is just a
rough sketch.
The domain,
that means the values of x,
was between negative infinity
and infinity.
The range, the y values,
they went back and forth
from negative 1 to positive 1.
And then we've easily defined
that this had a period
of 2 pi.
In other words,
before it started repeating
itself, it went 2 pi units
on the x axis.
Now, I've sort
of compressed this
so you could see that,
you know, this sine function,
or "sine wave,"
as it's called, you know,
has a lot of these little
humps here going above
and below and going
through 0,0 and then pi,
and then 2 pi,0 and 3 pi,0,
etc. So I'm not putting
in a lot of values,
and I'm just doing a rough
sketch of this.
Make sure that when you do it,
use a pencil, and, you know,
you kind of curve it.
Don't make it look
like it has any sharp edges
at all.
So what I have
at the bottom here is the
precise definition
for a periodic function.
A periodic function
with period P is a function f
such that f of x equal f
of x plus P
for every real number x
in the domain of f,
where P is the smallest number
for which this is true.
Man, that's hard to swallow
and understand.
But what it's saying here is
if you put a number x and,
like, if the period is 2 pi,
that means if I did f
of x plus 2 pi,
I would get the exact
same value.
And in this case,
that's actually true
because if you have a real
number and you add 2 pi to it
and you're
on that unit circle,
you're going to end
up in the exact same spot
with the same value
for your y coordinate.
All right.
So that's sort
of what this means.
But looking at it,
you get a better idea.
So if you start somewhere,
you just pick a place to start
and think where,
how long is it before it
starts repeating itself,
I might say okay,
let's say I started right here
at 0,0.
That's a nice place,
and I see it's going up
and down and up,
and then it starts
repeating itself.
So how long,
how big is that space?
Well, it goes from 0 to 2 pi,
and that's
where we get the idea
that the period is 2 pi.
Now, somebody else may have
not started right there.
They said, well,
how about if I start, like,
over here,
a little to the left of 0,0,
which is right here.
In other words, that,
on the x axis,
this here is negative pi,
right?
Here's pi.
There's negative pi.
So if I go in between here
and here, I see
that if I start here
at negative pi,0,
it goes down; up through 0,0;
up to here, which is pi
over 2,1; and then back
down through pi,0;
and then it starts repeating.
And that also is 2 pi units,
right?
Between negative pi and pi.
Well, this is pi units plus
another pi unit,
so that distance is 2 pi.
So visually,
you should be able to tell
that the period is 2 pi.
All right.
There's something else we want
to talk about,
and that's the amplitude.
And before going
on the definition, basically,
if you're looking
at the amplitude
of the sine function,
we're going to talk
about amplitudes
of only the sine
and the cosine function,
saying if you look
at the very bottom
and the very top,
it's half the distance,
is the amplitude.
So it is --
how far is it from the bottom
to the top?
Well, it goes from -1 to one.
That's a distance of 2 spaces,
and half of that distance
is 1.
So we would say this has an
amplitude of 1.
[writing sounds] It's a
positive number.
Some people call the amplitude
the height,
but then you might think oh,
it's 2 units high.
And actually it means
from the center,
how high is it,
not from the very bottom
to the very top.
It's only half that distance.
So here is a
precise definition.
The amplitude of a sine wave,
which could also be called the
"sine function,"
is the absolute value
of half the difference
between the maximum
and minimum y coordinates.
So if you take the maximum y
coordinate 1 minus a -1,
right?
So if you did 1 minus -1,
and then you divided by 2,
2 over 2 is 1.
So you can see how
that does make sense,
that you would get an
amplitude of 1, if you wanted
to use this
precise definition.
The other way is you just look
through the center here.
The center is really the x
axis here, right?
And it goes up 1,
or it goes down 1.
So the amplitude is defined
to be 1.
All right.
So this is your basic,
most basic, trig function:
y = sin x. It's the first one
everyone learns how to graph.
And you want
to get comfortable with this,
so that if somebody says graph
y = sin x,
you can go right to it.
There's always --
I remember it always starts
at 0,0, and it ends at 2 pi,0.
It goes up and then down
and back to 2 pi.
But, of course,
you could start it over here
at -2 pi and go up, down,
and ending right here.
That would be one period,
right?
So if somebody asks you
to graph one period,
maybe somebody's would look,
you know, just this part
that I'm going to do
in red right here.
Somebody might say okay,
that's one period.
Okay. That's true.
That's one period.
Somebody else might draw their
period like this.
Okay, I'm going to start here,
and I end up there.
Okay. That's fine.
That's one period, as well,
because it's 2 pi units
on the x axis.
But most people will do this
one, starting here and going
through here.
If I said two periods,
you'd have to go again, right?
But you want
to get the basic idea
of how this goes
and do a pretty good rough
sketch of it.
All right.
Here's the very important
thing to remember.
So if you were
to graph y = sine of x,
you want to know these five
key points, in other words,
these five key ordered pairs:
0,0; pi over 2,1; pi,0;
3 pi over 2,-1; and 2 pi,0.
So if I were going to do this,
you first of all want
to make sure that you mark
on your axis --
well, for instance,
let's say that this is pi.
Then we know every other hash
mark is going
to be pi more, right?
Here's negative pi and -2 pi.
So on the x axis,
you want to be consistent.
There is where a lot
of people have trouble
when they have, like,
pi over 3 and pi over 4,
and the markings aren't really
in the right place
on the x axis.
We'll go over that later.
And then, of course,
we want to be consistent
differently on the y axis.
Let's say we're going to go
in between -1 and 1.
So we want
to plot these five key points:
0,0; [writing sounds] pi
over 2,1; pi,0; 3 pi over 2,
1; and 2 pi.
And that will give us
one period.
So you always want
to mark these key points.
Now, you can also write the
ordered pairs in, but as long
as you've labeled what's
on the x and y axes,
it should be clear what those
ordered pairs are.
And what makes --
and this really keeps going on
and on, right?
Like, we could do some more
ordered pairs.
For instance, we can do 2.
But remember to put arrows,
and don't make it look
like it goes up forever,
necessarily, or down forever.
You know, it's going
to keep going like this.
So we could label these
ordered pairs, or,
as long as you're very clear,
marking on the x and y axes,
it should be clear.
But what makes them key points
is that we sort
of have a beginning
of a cycle.
We have the maximum point
of the cycle.
That's here at pi over 2,1
for y = sine of x,
and we have the center
of the cycle right here
at pi,0.
We have the minimum,
one of the minimum points
of the cycle right here
at 3 pi over 2,
1 and sort of where it ends.
So that's how the five key
points are defined.
You always want to have
at least five key points
for a sine function.
So you want to remember this:
It starts off at 0,0;
ends up at 2 pi,0.
It goes 0, 1, 0, -1, 0.
That is for y = sine
of x. That's going
to help us graph other
functions in --
with the sine, for instance,
t sine x or -5 sine x, etc.,
which we'll be doing on some
of the next few videos.
This is your Math Gal
Julie Harland.
Please visit my website
at YourMathGal.com,
where all of my videos are
organized by topic.