>> Julie Harland:
This is YourMathGal
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.
All right,
we're continuing graphing trig
functions, still working
on the sine function here
and in this video we're going
to do this problem right here.
So we need to find the period,
the amplitude,
and the phase shift
and then graph.
So we're getting
into a more
complicated problem.
The amplitude is
pretty simple.
This one-third up here
so that one's going
to be simple but we also need
to state what the period is
and we need to,
let's do the amplitude
and we're going to have
to do the phase shift.
Okay, the amplitude is the
easy one.
That's one-third
because that is the
coefficient in front
of the sign.
And of course this is absolute
value so if it was negative
one-third the amplitude would
still be one-third.
All right,
now what about the period
and the phase shift?
Okay, the period you can also
get by taking the coefficient
of X and that is your B,
so to get the period note B
equals 2 because that comes
from right here.
And so to get the period we
always do 2 pi over B,
2 pi over B, so in this case
that would be 2 pi over 2
which is pi units.
So the period will be pi.
It doesn't matter
if something is added
or subtracted to it
or if there is a parenthesis
next to the two it's the
coefficient
of X that's what your value
of B is.
And instead
of writing equal I'm just
going to put, there,
all right,
so we've got a period of pi.
Now what about the
phase shift?
So if you remember
from the problems
where we were just doing
problems with a phase shift
like Y equals sine
of 2X what we did was you
would take whatever is
in the parenthesis,
the whole thing
in the parenthesis
and say how would we,
what would that equal?
What will X have to equal
to make that zero?
So basically, you'd take that
and set it equal to zero.
So for the phase shift
remember, you're going
to say 2X plus pi is equal
to zero and then solve for X.
So if X was negative pi
over 2 it would make what's
in the parenthesis,
it would make the argument
inside here zero.
So the phase shift well,
the negative sign tells us
it's going to the left.
So this will be pi
over 2 units to the left.
I'm not going
to write the word units just
because I don't have a lot
of space right here
but everything gets shifted pi
over 2 units to the left.
So we've got to period,
the amplitude,
and the phase shift.
Now we're going to need
to find our key points
starting with the
X coordinates.
So here is our equation.
We know the period,
the amplitude,
and the phase shift.
Well, what does this mean
if the phase shift is pi
over 2 to the left?
Remember when we solved it 2X
plus pi equals zero,
we ended up solving it
and getting negative pi
over 2.
This will be our first X
coordinate
because everything got shifted
over pi over 2 units
to the left and so
that is actually the first
ordered pair we want.
Instead of zero we're moving
to the left pi over 2
which is negative pi over 2.
Now we need
to get the rest of these.
I think I'm going to have
to write this like this
to get it to fit.
Okay there we go.
So then how would I get the
next X coordinate?
So if you start at negative pi
over 2 and we've got a period
of pi remember we need
to add one-fourth
of that period
so that we're actually adding
one-fourth of pi,
so it's a way
of thinking about it.
Here's negative pi over 2
and you want to end
up pi units, right?
Oops, sorry.
You want to end up pi units
to the right.
So in other words
if I added pi to pi
over 2 I would end
up where I wanted but I need
to know these places
in between,
so I have to just first add a
fourth of pi
and then another fourth of pi
and then another fourth of pi.
So that's what's going
on here.
I start out with my first X
coordinate
and I add a fourth of pi.
Okay, now to add these I need
to get a common denominator.
So that would be negative pi
over 2 plus pi over 4.
So the common denominator is 2
so I need to multiply this
first one by 2 over 2
so that's really negative 2 pi
over 4 plus pi over 4.
Yes, you have to be good
at your fractions to do this.
So negative 2 pi plus pi gives
you a negative pi over 4
for your next coordinate.
Now it's easier if all
of your X coordinates have the
same denominator.
So let's rewrite this negative
pi over 2 with the same
denominator
which is right here.
So it will be easier
if we think of this
as negative 2 pi over 4.
Now how much am I adding
to get to the next place?
I just add pi over 4 again,
so I added pi over 4.
Here it is.
That's what I'm adding.
I'm always adding a fourth
of pie to get the next X
coordinate so if I'm
at negative pi over 4
and I add pi over 4
to that what is negative pi
over 4 plus pi over 4.
I'm now at zero.
So that's my next coordinate.
Now we're going to add pi
over 4 to that.
That will be pi over 4
and I'm going to add pi
over 4 one more time
that is 2 pi over 4
and now the question is,
is that a full pi unit?
So the question is notice I
started at negative 2 pi
over 4 or you could think
of that as negative pi
over 2, right?
And if the period is pi you're
adding pi and so
to add pi you're really
thinking it is 2 pi over 2
and that would give you
negative pi plus 2 pi.
Your final answer you should
end up at pi over 2
which is the same thing
as 2 pi over 4.
So hopefully that makes sense
that you would end up at 2 pi
over 4 which is the same thing
as pi over 2.
So that's how we get our
X coordinates.
Notice you just add pi over 4;
add pi over 4; add pi over 4;
add pi over 4 and that means
if you just had added the
entire period
to the first X coordinate you
should end
up at this fifth X coordinate
which is where the period end.
Now what about the Y values?
I think I'm going
to make all this just slightly
smaller because I'm running
out of room.
So I rearranged the page a
little bit
so you can still
see everything.
Now how about any new
Y values?
I've got one-third the sine
of this stuff
but remember the reason we
chose these values of X is
because those will give us the
usual values where we can
for X and sine X
and what really matters
are these.
Remember it goes zero, 1;
zero, negative one,
zero so I just get
to multiply one-third by all
of these values.
So I'll have one-third times
something,
all of these values get
multiplied by a third.
And I start multiplying
by zero then by one,
then by zero then
by negative one,
by zero of course you don't
have to show that step.
You could that in your head
but you do have
to know these values
over here.
And remember,
that's for the usual values we
have for sine and again,
keep in mind
by using these values we have
for the X coordinates here it
will be the same after we plug
in those values
for X. X is just taking the
sine of the Xs you see here.
It's a little confusing.
Okay so, one-third
of zero is zero.
Let me make those in purple.
One-third of zero is zero.
One-third of one is one-third.
One-third of zero is zero.
One-third of negative one is
negative one-third
and one-third of zero is zero.
So these are our four,
five ordered pairs we're going
to use to graph
and that is our last step.
By the way,
it usually takes the whole
page to do these problems
because this is a pretty
complicated problem so we have
to decide where to put our X
axis numeration system
on each.
Notice we're going
from negative 2 pi over 4
to 2 pi over 4.
So about the same to the left
as to the right,
so every four spaces is
pi right?
Our increments were
in fourths.
So I've got one, two,
three, four.
That's where pi would be.
One, two, three, four.
That's where negative pi
would be.
Would about on the Y axis?
We're going
in between negative one-third
and one-third.
So how about just calling
this one-third?
And we'll call
that negative one-third.
If that bothers you instead
just go up
and say there's one how
about that?
It's not very consistent.
Let's do it one more time.
One, two, three;
call that one.
Call that negative one.
All right so we've got our
markings we're marking very
carefully, we're very careful
to mark these,
so I have negative 2 pi
over 4 is zero.
So that means I'm going
over two spaces to the left
or negative 2 pi
over 4 and zero.
Then we've got negative one pi
over 4 up one-third.
There's up one-third,
zero, zero.
Pi over 4 negative one-third
and 2 pi over 4,
zero and there we have you can
hopefully see the shape
and if I keep going I could
also do these points
and there we go sketching our
sine curve
and make sure the arrow is
at the end.
All right, so there we are.
Y equals one-third sine
of 2X plus pi
and as always it's a good idea
to plug one of these numbers
in and make sure
like for instance,
make sure that that really is
a correct ordered pair
by plugging in pi over 4 for X
into here.
So you know
that Y should be one-third.
So this is just a check.
Y equals one-third
of sine 2 times what am I
plugging in?
X over 4 plus pi.
And then I'm actually going
to do that computation.
So that's one-third
of sine what?
Right. So this is 2X over 4
which I'm going to write as X
over 2 and getting a common
denominator,
I'm sorry that was supposed
to be pi over 4 not X over 4.
Pi over 4.
All right,
so getting a common
denominator that's pi
over 2 plus I'm getting a
common denominator here
and write that as 2 pi over 2
so that really gives me keep
on going one-third sine
of what?
Well you've got pi
over 2 plus 2 pi
over 2 that's 3 pi over 2,
3 pi over 2.
And then what's the sine
of 3 pi over 2?
Well, you know
that is negative one
and that's negative one-third.
So when X was pi over 4,
Y was negative one-third, yes.
When X is pi over 4,
Y is negative one-third.
A lot of people will not take
the time to do that
but it's easy
to make a mistake
and do something wrong
when you fill out your chart
so I really recommend you
at least do one check
like that.
And you just don't assume
that you have all the correct
ordered pairs.
All right now remember we had
a phase shift of pi over 2
to the left
and this is the graph
of just it looks
like you see the sine
curve here.
It looks like everything got
shifted over pi over 2 units.
So this is kind of easy to see
and the amplitude
was one-third.
Notice it is just up one-third
and down one-third
and the period you can see is
in between zero and 2 pi.
So hopefully it all
makes sense.
This is YourMathGal
Julie Harland.
Please visit my website
at YouMathGal.com where all
of my videos are organized
by topic.