>> This is Your Math Gal
Julie Harland.
Please visit my website
at yourmathgal.com where all
of my videos are organized
by topic.
All right.
In this video I'm going
to help show you how
to find the find the five key
x coordinates
for graphing the cosine
or the sine function.
So I have here the charts
for y equals sine of x
and y equals cosine of x,
and you should be familiar
with these.
For one thing,
you note that the period
for the basic sine
or cosine function is 2 pi.
So notice for the x
coordinates, if it starts at 0
and the last key coordinate is
2 pi, that should be the total
of 2 pi unit.
So the difference between 2 pi
and 0, for instance,
is 2 pi units.
Now, to get each x coordinate,
whatever the first x
coordinate is, we're going
to go a fourth of the way
from our first key coordinate
to the last one.
So if we have 0 and we want
to end up at 2 pi, we --
remember we need
to find these other ordered
pairs in between it.
All right?
So one way of doing
that is you're going
to just add a fourth
of the period.
So you're going
to add a fourth of 2 pi
to get this new ordered --
this new x coordinate.
Okay. So how about we draw
that down here.
Note: One-fourth
of the period is 2 pi.
I'm sorry.
One-fourth
of 2 pi is pi over 2.
So one-fourth
of the period tells us the
space between the first x
coordinate
and the second x coordinate
of our key points.
So we have 0, we're just going
to add pi over 2 to 0
and that is why the second x
coordinate is pi over 2.
All right?
So in other words,
I just added pi over 2, right,
to 0, right.
That's how I got it.
Now, how would I get the
next one?
I'm at pi over 2,
I add pi over 2 again.
I add one-fourth
of the period.
All right, you could write
that as 2 pi over 2 or pi.
Okay. Now,
how would I get the next one?
I'm going to keep it as 2 pi
over 2 here,
and I add pi over 2.
I again add one-fourth
of the period
and that's my 3 pi over 2.
And if I do it one more time,
if I take 3 pi over 2
and add a fourth
of the period,
I get 4 pi over 2
which is 2 pi.
So that is how we get the
x coordinates.
If we know
where we're starting --
in the case of y equals sine
of x and the y equals cosine
of x, we're starting at 0 --
we start with our first x
coordinate,
we figure out the period,
take one-fourth of the period
which is pi over 2,
and add that to the
first coordinate.
So we start at 0 for both sine
of x and cosine of x,
start at 0
for your first x coordinate,
add pi over 2,
add pi over 2 again,
add pi 2 over --
over 2 again,
and add pi over 2 again.
Now, you can write these
with a common denominator,
and actually, I suggest that's
in most cases.
So if we add pi over 2 to 0,
we get pi over 2; if we add pi
over 2 again, we have 2 pi
over 2; add pi over 2 again,
we have 3 pi over 2;
and then we've got 4 pi
over 2.
Okay. So it's
up to you whether you want
to write it
with a common denominator
or not.
We'll find out that
in most cases,
if there's a phase shift
especially,
you're better off writing
everything
with a common denominator.
All right.
So let's try this one.
Figure out the five key x
coordinates
for y equals the cosine
of 3x over 4.
So there is no phase shift.
Remember, the phase shift
there has to be
like plus something
or minus something inside
the parentheses.
So we would just say the phase
shift is 0,
or you would say there's no
phase shift.
But what about the period?
Okay. So let's do the period.
The period is 2 pi divided
by b. So in this case,
b is three-fourths.
Okay. Note:
b is three-fourths.
So I'm going
to do 2 pi divided
by three-fourths,
which is 2 pi times
four-thirds,
and we've got 8 pi over 3.
So that is the period.
So now we need to figure
out what's one-fourth
of the period.
Well, it's one-fourth
of 8 pi over 3.
And let's see.
Let's cancel,
the 4 goes into the 8 twice,
and so it's 2 pi over 3.
That makes sense.
One-fourth of 8 pi
over 3 would be 2 pi over 3.
Hopefully,
that makes sense to you.
Now, if you don't have a phase
shift, then our first x
coordinate is at 0.
So if your first one is at 0,
okay, that is your second,
because 2 pi
over 3 is your second x
coordinate,
because you'll be adding it
to 0.
So let's write
down what we know.
Let's just write the first x
coordinate,
the second x coordinate,
etc. All right.
So what would the first
one be?
Since there was no phase
shift, or 0 phase shift,
it will be 0.
The second x coordinate will
be just simply a fourth
of the period
because you're adding it
to the 0.
So I'm going to write down how
to get it.
It's really you're taking the
previous x coordinate
and adding one-fourth
of the period,
which is 2 pi over 3.
Third x coordinate;
what would you do?
You would take the 2 pi
over 3 and, again,
you would add a fourth
of the period,
that gives me 4 pi over 3.
Fourth
x coordinate,
you would take what I have
so far, 4 pi over 3, and,
again, keep adding one-fourth
of the period,
so that gives us 6 pi over 3.
And then the fifth x
coordinate, we would take
that 6 pi over 3 --
let's see --
and, again, add --
you keep adding 2 pi over 3.
You keep adding a fourth
of the period.
This gives me 8 pi over 3.
Now, we know
that the period was 8 pi
over 3, right?
Whoops. Sorry.
That was the period.
So I started at 0,
and my last x coordinate is
at 8 pie over 3.
So these will be our five
x coordinates.
So let's put those in a chart.
And the last thing I would say
is, before you put them
in the chart --
in a chart,
make them all the
same denominator.
Of course,
it's okay to just write 0.
So if some
of these x coordinates had a
six in the denominator,
then I would just put
everything in terms of 6.
What's nice is here they all
happen to have a
common denominator.
So let's make a chart now.
All right.
So remember, we're trying
to get a chart
for y equals the cosine
of 3x over 4.
And these are now how we get
the five key coordinates.
So for the cosine, 3x over 4.
How would we get what --
those are the y coordinates
we want.
So since it's just simply the
cosine of something,
you could think
of this whole thing
as the y values
in the basic chart
for cosine x. So remember,
in the basic chart,
it starts at 1
for the cosine, right?
So we start at 0, you know,
remember, it's the usual
suspects here.
Whoops. Sorry.
Technical difficulties.
Okay, here we go.
Pi, 3 pi over 2, and 2 pi.
Remember how it goes, 1, 0,
negative 1,
0 and back up to 1;
these are the y values
for the cosine of 3x over 4.
So we're going
to just put those in 1, 0,
negative 1, 0, 1.
This is Your Math Gal
Julie Harland.
Please visit my website
at yourmathgal.
Com where all
of my videos are organized
by topic.