>> This is your Math Gal
Julie Harland.
Please visit my website
at YourMathGal.com
where all my videos are
organized by topic.
This is Part 7
of graphing trig functions.
We're going
to be doing horizontal
translations
of the sin function.
So we're still working
with the sin.
And we're going to do problems
of this form, y equals the sin
of x plus c. So remember
when we graph y equals sin
of x we had these five key
ordered pairs 0 0 pi over 2 1,
pi 0 3 pi over 2 negative 1
and 2 pi 0.
Now, we want
to graph y equals sin
of x plus pi over 4.
All right.
So here's the trick.
This time in parentheses --
so what are we taking the
sin of?
Is x plus pi over 4.
Now, we look at this chart
for the x equal sin of x
if that were 0.
In other words, if x plus pi
over 4 was 0 we know
that the sin would be 0.
If it was in the parentheses
was that pi
over 2 we know the sin would
be 1.
So the trick is to find values
of x so that the sin will be 0
1 0 negative [inaudible] 0
as it is for y equal sin of x.
So we need
to choose the right ordered --
the ordered pairs here.
And that means we got
to find --
we have to find the simplest
values for the x coordinates.
Okay. So I'm going
to start off by saying, well,
what would make that 0?
So whatever is
in parentheses --
you said it equaled
0 [silence].
And we get x equals negative
pi over 4.
So look, what
if I had put negative pi
over 4 in for x
in these parentheses I would
now be able
to get the sin of 0.
So this is telling me that one
of the ordered pairs are here,
right?
If you put in negative pi
over 4, you'll get 0, right?
Look at this value over here.
So if I were going
to fill a chart in instead
of being 0 0 I've got negative
pi over 4 0.
Now, notice I could taken
that x equals negative pi
over 4.
Basically what I did was
really just subtract pi over 4
from this 0
to get this negative pi
over 4, and that's how we're
going to get our five
key points.
We're going to subtract pi
over 4 from all
of the x coordinates here.
All right.
So if I do pi
over 2 minus x minus 4 --
and here's where you have
to be very careful
with your fractions
and arithmetic.
So in other words,
I want the x coordinate pi
over 2 minus pi over 4.
They need a common
denominator,
and that would be 2 pi
over 4 minus pi over 4
which equals just pi over 4.
Oops. I don't know why
that does that.
Sorry. So here's another
ordered pair.
I'm going to choose pi over 4
for my x coordinate
and what happens?
If you put pi over 4
in here -- why do I want this?
Pi over 4 plus pi
over 4, right?
Would add up to 2 pi over 4
which is pi over 2.
Huh. And then I would take the
sin of pi over 2
which I've got in my chart
over here that would give me
the ordered pair of 1.
Now, I could continue
with this x coordinate.
I'm going to subtract a pi
over 4.
So I take pi over 1
and minus pi over 4.
And I need a
common denominator.
So I need to multiply this
by 4 over 4.
Now, I got 4 pi minus pi
in the numerator that's going
to give me 3 pi over 4.
And when you plug in 3 pi
over 4 for x what happens?
Look up here in the yellow.
You would have 3 pi
over 4 plus pi
over 4 that's 4 pi
over 4 or pi.
And again, if we take the sin
of pi we're going get 0.
[Silence]
We could do the same thing we
could take 3 pi over 2
and subtract pi over 4.
And then we're going
to take 2 pi
and subtract pi over 4.
But I want
to show you an easier way
to do that.
The x coordinates go
up by the same amount.
Look over here for y equal sin
of x. I'm going up by pi
over 2, right?
I'm first adding pi over 2.
I'm adding pi over 2 again
to get pi.
I'm adding pi over 2 again
to get 3 pi over 2.
And I'm adding pi over 2 again
to get 2 pi.
How much am I adding
from negative pi over 4 to get
to pi over 4?
I'm adding 2 pi over 4.
And then I'm adding 2 pi
over 4 again
to get 3 pi over 4.
So if I add 2 pi over 4 again,
that should be my next value.
For x, and then if I add 2 pi
over 4 again I should get 7 pi
over 4.
That would be the fifth
key point.
Now, you'll get the same
answer if you do 3 pi over 2
and subtract pi over 4.
All right.
So just to make sure you
believe that --
3 pi over 2 minus pi
over 4 is what?
6 pi minus 1 pi.
See how I'm going to get
that 5 pi over 4.
So you could either take each
of the x coordinates
and go ahead and subtract --
sorry -- this from each
of the x coordinates.
That's what I'm doing
to get these new key points
for x, okay?
And then our y values will be
the ones you've already see
here under sin of x.
So what you're doing is
creating some x coordinates
so it's really easy
to compute the y coordinates.
So the y coordinates will be
the 0 1 0 negative 1 and 0.
Now, it is easy
to make a mistake
so you probably should plug
in one of these numbers
and make sure it's true --
one of these ordered pairs.
So if I put in 5 pi over 4
for x in this original
equation --
if you put in 5 pi
over 4 that'll give you 6 pi
over 4, right?
When you add the pi over 4,
and 6 pi over 4 is the same
thing as 3 pi over 2.
And the sin at 3 pi
over 2 will be negative 1,
and that's exactly what I've
written --
5 pi over 4 negative 1.
So these are five key points.
Now, you don't have
to write them
as ordered pairs you can make
a little t chart just
like I did over here.
The trick is how are you going
to get these x coordinates?
There's two ways.
Subtract pi over 4
from the each
of these x coordinates, okay?
Or just do it
for the first one
which is always easy
because if you just do 0 minus
pi over 4 it's the exact same
thing, right?
And then you're going
to just subtract it
from this second and then say,
well, what's the
difference here?
I'll go up by 2 pi, right?
I'm going from a negative 1
to a positive 1 that's
in the numerator.
Okay. Now to graph this.
What's very important is
that you always have your x
coordinates having the same
denominator that'll make it
much easier.
And notice I've got them all
in fourths.
So I want to draw a graph
where everything is a fourth
of pi.
All right.
You want to go in increments,
you know, you want to go
in equal increments
on the x axis.
All right.
So we're going
to graph y equal sin
of x plus pi.
Before we came
up with these five key points,
and now just writing it
in a chart as opposed
to ordered pairs.
And so we need
to make increments in terms
of a fourth of pi --
pi over 4 means a fourth
of pi.
So that means every fourth
mark will be pi, right?
That's 1 pi over 4,
2 pi over 4, 3 pi over 4,
4 pi over 4,
this would be 5 pi over 4,
6 pi over 4, 7 pi over 4,
8 pi over 4 would be 2 pi,
9 pi over 4 etc.
and on the other side.
[Silence]
Negative 1 pi
over negative 2 pi
over 4 negative 3 pi
over 4 negative 4 pi
over 4 is negative pi.
Very important
that you make this as accurate
as possible.
And then the highest
and lowest points here are
going to be negative 1 to 1
so I'm just going
to label this 1
and this one negative 1.
And here we are.
So let's graph.
This one's [inaudible]
of y equals sin of x plus pi
over 4 by plotting
these points.
So we have negative pi
over 4 0.
All right.
So the x we go to negative pi
over 4 0, and then we've got
pi over 4 1.
So here's pi over 4 up at 1.
We have 3 pi over 4 0.
So 1 2 3 pi over 4 is 0,
and then we have 5 pi over 4.
Notice I just keep going two
increments to the right
to get my next ordered pair
maybe it's up or down.
But here's two increments more
to get to 5 pi over 4,
and that's down at negative 1.
And then 7 pi
over 4 is right here.
It should be very consistent.
And you should see that shape
of the sin curve
that you're getting used.
And of course,
you can keep going like.
If I went to the left two I
should be down here
at negative 1 again.
And then if I go up,
I should be at 0 again etc.
So I could see a little bit --
a few more ordered pairs here.
If I go two to the right,
I should be up again,
etc. So let's graph this.
So we want to curve it as much
as possible.
Make sure you don't put any
sharp curves in it.
Make it a sketch
like you're an artist here.
All right.
And here I'm just going
to keep going a little
bit further.
All right.
So I think I've graphed almost
two cycles of this function.
Yeah. Let's just graph y
equals sin x in green, okay?
Now remember how that looked?
We've got 0 0,
and then we had pi over 2 1,
and then we had pi 0.
3 pi over 2 negative 1
and 2 pi.
And of course I can go,
you know, a few more ordered
pairs here.
[Silence]
So it looks like if you look
at regular function y equals
sin of x this sin of x plus pi
over 4 is shifted to the left.
And how much has it shifted
to the left
from let's say this ordered
pair of 0 0
on y equals sin axis?
And here's this ordered pair
of negative pi over 4 0
on y equals sin
of x plus pi over 4.
The entire graph has shifted
pi over 4 spaces to the left.
Okay. So this all ordered
pairs --
[Silence]
On y equals sin of x
[Silence]
Yikes.
[Silence]
Has a mind of its own.
Okay. Here we go.
Y equal sin
of x are shifted pi
over 4 units to the left.
So how some people do this is
when they solve the argument
the part inside the ordered
pair, I mean,
inside the parentheses x plus
pi over [inaudible] remember
how we started?
X equals negative pi over 4.
Is that what we got?
This negative sin tells you
there's going to be a shift
to the left.
Minus sin means shift.
[Silence]
Shift pi over 4 units
to the left, okay?
So the left that's
because it was minus.
If it was positive,
we would actually be shifting
to the right.
And we actually call this
shift -- instead
of calling it a horizontal
shift when you're talking
about trig you say this is a
phase shift.
[Silence]
Of pi over 4 units
to the left.
[Silence]
Okay. So what does this mean?
Okay. So let's look
at how we started.
So here's our graph we
started with.
Y equals sin
of x plus pi over 4.
The period was 2 pi as usual.
The amplitude was 1.
The domain were all real
numbers you could fit anything
you wanted
for x. The range it went
between negative 1 and 1.
And the phase shift this isn't
something new.
Pi over 4 units to the left.
Remember what you do?
You take what's
in the parentheses solve it.
So if you solve x plus pi
over 4 equals 0 you get x
equals figurative pi over 4.
So the minus sin tells you
it's to the left
that many spaces.
This is your Math Gal
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.