>> Julie Harland:
This is YourMathGal
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.
We're continuing
to graph trig functions.
We're still working
on the sine and we're going
to do some problems
where we need
to determine the period
for the sine function
and choose the five key X
coordinates
which will help us get the
five key points
and then we'll graph.
All right,
so it says find the period
of Y equal sign of 3X
and state five key points
and graph.
All right, now there a couple
of ways of getting period.
You can use the formula
which is you take the
coefficient of X
which is 3 can and that's V.
So we have V equals 3
and the period
for sine will be 2 pi over B.
So in this particular case it
will be 2 pi over 3.
So that would be the period
for this graph.
The other way
to do it is you know
that you can just put 3X
in between zero and 2 pi
because 2 pi is the usual
period for Y equals sine X
and then if you divide
everything by 3 you'd solve
for X and so that tells you
that you have a period
between zero and 2 pi over 3.
So again you'll get a period
of exactly 2 pi over 3.
Now down here we have the
sine function.
Y equals sine of X and it goes
from zero to 2 pi.
Now this is going
to be much shorter.
It's only going to go
in between zero
and 2 pi over 3.
So the trick is figuring
out how you are going
to mark your X axis using
that information.
So here is the idea.
I'm not going
to draw the whole X and Y axis
but let's just think
about the X axis.
So you know the period starts
at zero and it ends over here.
You know, one cycle ends
at 2 pi over 3.
You need to have five
key points.
That's the first key point
and the last key point
so I need the one right
in between,
right in between here,
right in between here.
So there's a couple of ways
of doing this.
This is the way I think is
the easiest.
If you get a fourth
of the way there
and then another fourth
of the way
and then another fourth
of the way and then
to the end you've got your
key points.
Since you're starting
at zero a fourth of the way
to there is simply a fourth
of 2 pi over 3.
All right, so what is a fourth
of 2 pi over 3?
Well, let's see you can
cancel here.
And that gives you pi over 6.
All right,
so this should be pi over 6.
Now so you've gone zero to pi
over 6 for your first key
X coordinate.
So you have to go pi
over 6 again,
well just add that.
That's going
to be 2 pi over 6.
The third one is going
to be 3 pi over 6
and the fourth one is going
to be of course 4 pi over 6.
But that is the same thing
as 2 pi over 3.
Again, I'm just writing them
all with a common denominator
because it will be easier
to graph if we think
of it that way.
All right, so again,
if you figure
out what one-fourth
of the way is that's just
one-fourth times that period.
One-fourth of 2 pi over 3,
you're going
to get the first mark
and then you just add
that again
to get the second mark.
Add that amount again
to get the third mark and add
that again and you should end
up at that last spot
at 2 pi over 3.
All right,
given that now we're going
to the draw the X axis
and the Y axis but we're going
to use pi over 6 for each
of our marks on the X axis.
All right, let's see,
I forgot to write original
up here.
It's the sine of 3 X.
That was our problem.
Y equals sine of 3 X. We found
out that the period was 2 pi
over 3.
We came up with these key
X coordinates.
Now what are the
Y coordinates?
If we're plugging
in these X coordinates we
should get the normal Y
coordinates zero, one; zero,
negative one; zero.
And you could verify that,
for instance put in pi over 6
into this equation,
the sine of 3 times pi
over 6 you'll get the sine
of pi over 2 which is one.
So you could do
that for all of those.
But if we're changing these
ordered pairs you should be
able to get those five key
points that we normally get
on the Y axis.
So now we're ready
to graph the function.
All right, I've got it going
from zero to 4 pi over 6
or 2 pi over 3.
It doesn't matter
so let's say I just kind
of have X axis over here
and notice there's nothing
in front of the sign
so we know the amplitude is
still one,
so we know we're just going
to go up one space
and down one space.
Here for our markings we can
do, well, we've got it
in terms of 6.
One, two, three, four,
five, six, six.
That's where pi goes.
And you can fill in the rest
of these if you want.
You could say that's pi over 2
or you could say this is pi
over 6, 2 pi over 6,
3 pi over 6, 4 pi over 6,
five pi over 6.
It does get a little bit
crowded here
when you start graphing.
So let's do it.
We've got zero, zero.
Then we have pi over 6 once,
so we go one over, one up;
then 2 pi over 6, zero;
then 3 pi over 6,
negative one;
then 4 pi over 6, zero;
and there we have a
complete cycle.
If we kept going we would now
go over one and up one again.
Go over another space we'd be
down at zero again,
go over another space you'd be
down, oops
down here, etcetera.
Same thing if I were to move
over to the lowest.
So I know it's going
to look something like this,
sketch it very carefully
so it's nice and curvy,
whoops, and there we have the
graph of Y equals sine of 3X.
And what's important is you're
markings are carefully drawn
on the X axis.
So if you want you can say
that this is pi over 6.
There's room to put the pi
over 6 there
so that might be helpful.
This is YourMathGal
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.