>> 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 graphing trig
functions 3.
so far we have graphed Y
equals the sine of X.
And this is what we ended
with on the last video.
Y equals sine
of X. I've got these 5 key
points here
that you pretty much want
to memorize.
And this is 2 periods
of this function.
The domain goes
from negative infinity
to infinity.
There are no restrictions
on the X variable.
The range the Y value goes
between negative 1 and 1.
The period is 2 pi.
So you have a full cycle
between 0 and 2 pi
for instance.
And the amplitude is 1.
Remember that's half the
distance from the minimum
to the maximum point
of the graph.
So even though it's 2 spaces,
we call the amplitude this
height right here starting
from the X axis up,
or the X axis
down in this particular case.
All right,
let's graph a different
function now.
What would it look
like if I wanted to graph,
instead of Y equals sine of X,
Y equals let's do it
like this Y equals 2 sine
of X. All right?
So my Y value instead
of being the sine of X,
would be 2 times the sine
of X. So, now what does
that mean?
If I already know what the
sine of X is,
I would take that's what my
original function was Y equals
sine of X. 2 sine
of X would mean all those Y
values right here
in this second column here,
times 2.
So, if you put in 0 for X
for instance,
you'd have 2 times the sine
of 0.
2 times 0 is going to be 0.
But instead
of redoing it each time.
I already know the sine
of X is 0, right?
So 2 times the sine
of X would just be 2 times
that 0.
Similarly, when you put in pi
over 2 for X,
you would the sine of pi
over 2 you know it's one,
and then we multiply it by 2.
So basically I got
to multiply these values right
here in this column by 2.
So I would have these
5 values.
And this gives me the
key points.
But you have to remember,
you've got your X
and your T sine of X.
So the ordered pairs here are
going to be 0,0 pi
over 2,2 pi 0,3 pi
over 2 negative 2 and 2 pi 0.
All right?
Those are five key points.
So let's graph those.
0,0's right here,
same as the other one.
That's one key point.
Pi over 2,2.
All right, so here's pi
over 2, but we have to go
up to 2 instead of 1.
We have pi is 0.
so that goes to the same point
as a local sine X. 3 pi
over T negative 2.
I see I just labeled
this wrong.
That's negative 2.
And 2 pi 0.
All right,
so what happens is I get
this graph.
And that is a full cycle.
It looks just
like the sine function except
it's sort of stretched.
Its amplitude is different,
as you can see.
So if I kept going,
this is when it starts
repeating itself.
This would go up again,
this would go down again.
Something like that.
Again, I'm just doing a rough
sketch here.
This would go all the way
down to negative 2,
back up and down.
So, in blue,
they have Y equals 2 sine
of X. Now what's the period?
The period still is
in between 0 and 2 pi.
That's a full cycle.
So let's write that.
The period is still 2 pi.
But what's the amplitude?
The amplitude is 2.
Remember it's the difference
between the minimum
and the maximum value,
and then you take half
of that.
So that's four units total,
right?
So the amplitude is 2.
Let's also write
down for this one
with the domain
and range let's just write.
The domain
for the sine function is
always going
to be all real numbers.
So let's just write
down that domain
for all of these.
There's no restrictions
on the X. Okay?
But if I were
to write the range
down for each of these,
the range for Y equals sine X,
it goes in between negative 1
and 1.
But the range
for Y equals T sine X,
where is that?
The minimum
to the maximum it goes
between negative 2 and 2.
So the range here
and I'm going to have
to make this a little bit
smaller so we can fit this in,
okay?
Here we go.
A little more space.
So the range
of Y equals 2X would be
in between negative 2 and 2.
So now we have Y equal sine
of X. Now we've graphed Y
equals 2 sine of X.
So let me ask you something.
How would you graph Y equals 3
sine X. All right,
so let's say.
How would I graph Y equals 3
sine X?
So try that.
Put the video on pause.
Graph it either
on the same one here
or on your own.
You know, brand new one.
State the period.
State the amplitude.
And state the range.
And after you've done that,
take the video off pause
and see how you did.
All right.
Now if we do 3 sine X,
what's going to happen?
Well, if you know what the
sine of X over here, again,
this is the main thing you
always want to the remember.
Because between 0 and 2 pi X,
it goes 0,1; 0, negative 1, 0.
So all those values get
multiplied by 3.
And put it in.
0, 3, 0, negative 3 and 0.
So our key points here are 0,0
pi over 2, 3.
Pi, 3 pi over 2,
negative 3 and up.
And what's happening is
that this gets stretched
even more.
And there we have a little bit
more than one cycle.
If you were going
to do 2 cycles you can,
you know, go
down to here, go up.
Okay. I know I'm not doing a
great graph here,
but you get the idea.
So we have 3 Y equals 3 sine
of X. The period is the same.
So for Y equals 3 sine X,
the period sorry.
This keeps on coming down.
I'm going to move this graph
down just a little bit.
All right,
so what would the period
be now?
The period
of this is still 2 pi.
Right? There's one full cycle
between 0 and 2 pi.
The amplitude all right,
what's the amplitude
of this one?
Is 3. And what's the range?
It goes between negative 3
and 3.
And remember for all
of them sine X all
of these the domain is all
real numbers.
All right so let's see
if we could make a prediction.
All right so,
so far we've done problems
like this, Y equals A sine X
of X. I did just sine X,
which means A was 1.
I did when A was 2.
I did when A was 3.
So those are all
positive numbers.
So let's just assume for now
that coefficient is positive.
What would the domain be?
What are the values
that X could take on?
Well, negative infinity
to infinity.
It could be anything.
Right? The range
what happened?
The range went
if it was 2 wait, sorry.
Technical difficulties here.
All right, so if this was 2,
remember a number between 2
and negative 2.
If it was 3 it was
in between 3 and negative 3.
So the range is
in between negative A. Oops.
I'm writing infinity.
Negative A
and positive A. All right?
We know that A is positive
so this should make sense,
the period.
The period was 2 pi
for all of these.
And what was the amplitude?
It was simply A. We know
that's a positive number
because it says A is greater
than or equal to 0.
All right,
so this should help you
and you should be able
to get your key points
from this.
And we're going
to do some problems
where A is negative
on the next video.
And by the way,
if I had let A equals one
half, let's just think
about what
that would look like.
What if I wanted
to know what Y equals one half
sine X looks like.
Go ahead and put the video
on pause and see
if you can figure that out
on your own.
What would happen then,
is that the periods the same,
right?
But the amplitude would
be what?
The amplitude would be
one half.
So it would look almost
like this function,
but it would only go
up a half a space.
And there we go.
That's what your graph should
look like.
That's one cycle.
If you kept on going.
If you want a couple
of cycles,
you can go over here.
Now if you're not sure,
in any case,
how to graph something.
What you do is you put
in a lot of ordered pairs
and you connect the dots.
So if you don't remember,
you know, what's really going
on here to make it simple,
using these values.
And then if I do one half sine
X you could just plug in.
Put in 0 for X. Put in pi
over 2 for X. Put in pi.
Put in 3 pi over 2 and 2 pi,
and see if you get the
smooth curve.
You should get 0, one half,
0 and negative one half and 0.
So you can also have
fractions, it doesn't have
to be whole numbers.
This is from math gal,
Julie Harland.
Please visit my website
at YourMathGal.com
where all my videos are
organized by topic.