>> This is your Math Gal
Julie Harland.
Please visit my web site
at yourmathgal.com where all
of my videos are organized
by topic.
All right, we're continuing
in our graphing
of trig functions.
And on this video we're going
to graph this function
right here.
All right.
So we've got Y equals sine
of 2 X minus 3.
State the period, amplitude,
and vertical shift and graph.
All right, so let's write
out what we do know.
How do we get the period?
Well, there are a couple way
of getting the period.
One was just using
the formula.
Remember, the period is 2 Pi
divided by the coefficient
of X, all right?
So dividing by 2,
this tells us
that our period is Pi.
Right? So that's the formula
for getting the period.
Now sine usually goes
from 0 to 2 Pi.
What it's saying is it's going
to go from 0 to Pi.
So when we are choosing our X
coordinates, got the sine
of 2 X minus 3.
Right? We also want to figure
out what that will be.
We know normally we're going
from 0 to 2 Pi,
so if you just think about it
for a minute again, here's 0,
you want to end up at Pi
for this function.
So you're going to have
to have three other places
in the middle, right?
So this is a fourth of the way
to Pi then another fourth
of the way,
and then another fourth
of the way,
and then another fourth
of the way.
So we're going to start off
with 0 as usual, at 0,
but now we're going to have
to add a fourth
of the way to Pi.
Well, a fourth
of Pi is Pi over 4.
So to get the next one we
would add Pi over 4 again.
That would give us 2 Pi
over 4, which of course is the
same as Pi over 2,
I'm just writing it
with the same denominator,
and then the next place will
be 3 Pi over 4, right?
Just add Pi over 4 again.
And finally, you add 4 Pi
over 4, which is just Pi.
It's up to you whether you
want to write these
with common denominators
or not.
I think it's easier to think
of them that way.
So these are going
to be our key points
on the X axis.
Now remember the usual numbers
we remember
for just Y equals sine
X. Sorry.
We just have X and sine X
or 0, Pi over 2, Pi,
3 Pi over 2, and 2 Pi, right?
So notice all
of these are actually half
of what you see over there.
Basically,
that's just dividing by 2 Pi
over here,
and what about the --
what will the Y values
change from.
So usually it goes 0, 1, 0,
negative 1, 0.
What happens this time?
Well, if it was just the sine
of 2 X, since we changed our X
values, we would have just
been able to keep 0, 1, 0,
and negative 1.
But this minus 3 means I'm
going to have to subtract 3
from all these Y values.
Okay? So I'm going to have
to do instead of 0, 1,
negative 1, and 0,
I'm going to have
to subtract 3
from each of these.
Now you don't have
to show you,
you can do that in your head
if you want,
it's just 0 minus 3.
Just negative 3.
This will be negative 2.
Negative 3.
Negative 4.
And negative 3.
So our ordered pairs will be 0
negative 3,
Pi over 4 negative 2,
2 Pi over negative 3,
3 Pi over 4 negative 4
and Pi negative 3.
So let's graph this.
Key points.
We are going between 0 to Pi
to get one cycle.
I'll just go over here
and also go to negative Pi.
And what about my
Y coordinates?
It's ranging
between negative 4
and negative 2.
So I have got to make sure I
at least have space to go
down here to negative 4.
And so we've got our X
and Y axis labelled,
which is very important.
Now let's go ahead and plug
in these ordered pairs.
So we have 0 negative 3.
And we have Pi
over 4 negative 2.
So that will be -- oops.
I don't know why this does
this, sorry.
So we have Pi
over 4 negative 2.
That would be right here.
We've got Pi
over 2 negative 3.
So now remember,
Pi over 2 is the same
as 2 Pi over 4.
Right? So we were going down.
You can see what's going on,
remember, it goes up first
and starts coming down.
We've got 3 Pi over 4
and negative 4.
And we've got pie negative 3,
and you can see your
cycle here.
Now of course
if you kept going it would be
down here and back up,
and down, so if you wanted
to see two periods
and then sketch it nicely
so it looks nice and curved,
and put arrows at the end
so it shows you know this
really cycles and goes on
and on and on.
That's not a perfect-looking
sketch there,
but I hope you get the idea.
So if you were asked what the
first key points are,
you can write those
as ordered pairs, 3,
0 negative 3 Pi
over 4 negative 2, et cetera.
Sometimes you're asked
to label them right
on the graph.
So for instance, if you were
to label this point right
here, that's the 3 Pi
over negative 4.
It gets kind
of crowds, though.
Okay? You could also label the
X coordinates right here.
This is Pi over 4,
this is 2 Pi over 4,
or Pi over 2, if you want
to write it so it doesn't take
as much space up,
and 3 Pi over 4.
All right,
so that's how we're
graphing this.
We've got both a period change
and we didn't say what the
amplitude was,
but remember the amplitude,
the coefficient
of the sine was 1,
so the amplitude was easy,
in this case the amplitude was
still 1.
The absolute value
of the coefficient there.
And so the amplitude,
you can look at the pictures
of one, it's half the bottom
-- from the very bottom
to the very top,
the period with Pi,
and there was a vertical shift
and so that was the other
thing we wanted to say.
What's the vertical shift?
It was down 3 units.
So let's just write that down.
The vertical shift was
down 3 units.
Sorry these aren't all
in the same place.
Just trying to get it all
in one place.
So we've got the period as Pi,
amplitude as 1,
vertical shift.
And here's our graph.
This is Your Math Gal
Julie Harland.
Please visit my web site
at yourmathgal.com where all
of my videos are organized
by topic.