>> 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.
We're continuing just graphing
lots of trig functions.
There's so many
different types.
We're still
on the sine function
right here.
And we're going
to do this problem.
So we're asked
to determine the period,
amplitude, and vertical shift,
and then graph it.
Now, there's two ways
to determine the period.
One is you use the formula 2
pi over b. So remember,
the period is 2 pi over b,
so I have to know what b is,
and in this case,
you look at always the
coefficient of x,
which is three-halves.
Right? You have to think
of that as three-halves x.
So b is three-halves.
So 2 pi over b, remember,
is the same thing
as 2 pi divided by b,
which might be easier
since we're dividing
by a fraction.
So if I wanted
to find the period,
we could do 2 pi divided
three-halves,
which is 2 pi times two-thirds
-- because you multiply
by the reciprocal --
and we get 4 pi over 3.
So this is our period.
All right.
If you don't want
to use the formula,
what you say is, you say,
what's in the parentheses
here, the 3x over 2,
all right,
and the sine usually goes
in between 0 and 2 pi,
and what you do is you solve
for x to see what the period
of this goes between.
So I need to isolate x. I
multiply both sides
by two-thirds.
And that says I get x is
between 0 and four-thirds --
4 pi over 3.
So, again, if goes between 0
and 4 pi over 3,
I'm going to have my period
of four-thirds.
So no matter how we look
at it, our period is
four-thirds pi --
I mean, I'm sorry --
4 pi over 3.
Okay. So we've done that.
Now, we're also asked
to state the amplitude
and the vertical shift;
those are actually a little
bit easier.
Let's split this up.
Amplitude.
The coefficient
of the function sine is
negative one,
it's just the absolute value.
So the amplitude is still 1.
And the vertical shift.
All right, vertical shift.
We've got something plus 1.
So everything is shifted
up one unit.
So the vertical shift happens
to be up one unit.
So now we're ready
to find our key points
and graph.
All right.
So here's our function.
We want to get the x
and y values here.
And so how do we find the
x values?
We know it goes between 0
and pi, 4 pi over 3.
So we know it starts at 0
and ends at 4 pi over 3.
If you want to look
at the picture,
that means if you start here,
you end at 4 pi over 3
to get a complete cycle.
All right.
So the question is,
what are these points here?
And to get our first x
coordinate after 0,
that's one-fourth of the way
to 4 pi over 3.
So that's how I like to do it;
I just say it's one-fourth
of the way there,
so it's one-fourth of pi
over 3, and that is simply,
if you cancel the 4's,
pi over 3.
So our next x coordinate is pi
over 3.
And then we keep adding a
fourth more
to get our next spot.
So, for instance,
here's pi over 3.
What's the next place going
to be?
That's pi over 3 plus another
pi over 3 is 2 pi over 3;
pi over 3 again is 3 pi
over 3; pi
over 3 again is 4 pi over 3,
which is what I wanted.
And those are the same things
that go over here.
2 pi over 3; 3 pi over 3 --
now, I know that is pi.
If you want,
you could write pi --
and 4 pi over 3.
I'm just writing them all
with a common denominator.
Okay. So those are going
to be our key x coordinates.
Right? Now,
we need to find the
y coordinates.
Here's where we remember what
the usual values are for x
and sine x. And what's --
what we're really
concentrating
on right now are the y values.
Remember, it goes 0, 1, 0,
negative 1, 0.
And the point is,
by putting in these values
of x in here, we're going
to find --
we're really going
to be taking the sine
of these actual values
after you do three-halves
times it.
And so we're going
to do a negative times what
that -- those usual y values
over here,
negative times whatever
that sine is negative 3x
over 2, which we know what
they'll be; they are 0, 1, 0,
negative 1, 0.
Those are those y values
right here.
Right? And then,
what do we have to do?
We have to add 1, right,
to each of those.
So let's compute.
Negative 0 plus 1 is 1,
so my first ordered pair is
going to be 0, 1;
negative 1 plus 1 is 0;
negative 0 plus 1 is 1;
the opposite
of negative 1 is 1,
so I have 1 plus 1 is 2;
negative 0 plus 1 is 1 .
And so there are my five
ordered pairs that I want
to put on the graph.
So now we're ready to graph.
All right.
So on the x-axis,
to get a full cycle we have
to go between 0 and 4 pi
over 3, so that's what's
so cool about graphing.
You get to label on your x
and y-axis,
and you must label them.
So I'm going to go
on my x-axis 0 pi over 3,
2 pi over 3, 3 pi over 3,
4 pi over 3; there's going
to be four markings
until I get
to this 4 pi over 3.
Now, you can, if you want,
label this pi,
and then that's obvious,
this is pi over 3,
2 pi over 3, 3 pi over 3,
4 pi over 3;
but be very consistent here.
If we go to the left,
that would be negative pi.
So it's up to you
when you want to put each --
a marking at every place,
but you have
to at least mark one place.
And on the y-axis, we're going
in between --
what's the very smallest
number is 0 and up to 2.
So I don't have anything below
the x-axis.
It's only going up two spaces
at the most.
So let's graph these ordered
pairs: We have 0, 1;
pi over 3, 0.
Okay, so the first place,
right there.
Notice it's going down first.
Why is that happening?
Because of this minus sign
in front of the sine.
Right? So that's why you see
it, it starts going
down first instead of going
up like the regular sine
function does.
Then we have 2 pi over 3, 1,
so it's the next place
and going here.
Right? The we have 3 pi over 3
which is the same thing as pi,
up two spaces.
And 4 pi over 3, 1.
So this looks
like the sine function except
it's going down first.
Right? It's got
that mirror image.
Right? It's flipped because of
that minus sign.
And if we wanted
to keep going,
we could do some more ordered
pairs over here
to get two cycles.
And the last thing you want
to do is make sure
that you computed these right.
So you might actually plug
in one of these values for x
into the original function
just as a check,
because it's easy to,
you know, kind of go
on autopilot
and make some
arithmetic mistake.
So, for instance, if I wanted
to plug in for x, pi,
so I'm going to plug in,
for instance, 3 pi over 3,
which is the same thing as pi,
I would be putting
in 3, right.
You'd do a 3 pi over 2,
this is equal
to negative, plus 1.
All right,
and actually compute it.
So that's negative the sine
of 3 pi over 2.
So what's the sine
of 3 pi over 2?
Remember what that looks like.
It's down here.
At 0, negative 1,
the sine is negative 1.
So that's the opposite
of negative 1 plus 1,
which is 1 plus 1 or 2,
and that's exactly what
I have.
Pi 2 was a correct
ordered pair.
So this is just a little check
you might do.
It's up to you,
but I suggest that,
especially
if you're taking a test.
This is Your Math Gal
Julie Harland.
Please visit my website
at yourmathgal.com where all
of my videos are organized
by topic.