OK.
So what are phase shifts?
And the best way I want to do this is let's
look back at other functions and how we shifted them.
And I was helping some other students here
in my Algebra II program.
I was working with them and they're like,
oh, this isn't bad.
When are we going to use this?
And well, here's an example in trigonometry
where we're going to use it.
So let's just use a standard quadratic here.
So here, I have a parabola and the standard form equation
of this is y equals x-squared.
And then what we talked about is if I
was going to now say, well, what happens if I make a graph--
and actually, let's use a little color coordination here.
What about if I now said, y equals x minus 2 squared?
Well, by using our definition of transformations
and translations, we now know that my graph is going
to be shifted two to the right.
It's going to be the exact same graph,
just now transposed over two units to the right.
And if I was going to do another alteration with that graph,
which--
we're actually not going to work with this.
But let's say if I wanted to say, y equals x plus 1 squared,
well, what that meant is now I'm going to shift my graph--
exactly the same graph--
one unit to the right.
And these are what we call our horizontal shifts of a graph.
And we also have vertical shifts of a graph
but what we're concerned about right now is our phase shift.
How can we move the graph left or right?
And it was important for us to understand
why we did the opposite direction
because when we look at the standard form of a quadratic,
the standard form of a quadratic says, y equals a times
x minus h-squared plus k.
And this is very, very similar to our trigonometric function
we're going to use.
And notice that you can do the x opposite of h.
So therefore, when I say, x minus 2, that means I'm
going to be shifting the opposite to the right two
units.
So this is a little review, a good review for us
to understand our phase shifts.
So to understand the phase shifts,
let's take a look at our standard form of our sine
and cosine graphs because that's what
I want to focus this video on.
So I have y equals a times sine of bx minus c plus d.
And I didn't want to put the sine in there.
And let's do y equals a times cosine of bx minus c plus d.
So now it's very important for us
to understand that we have that minus c again.
Just like the x minus h, we have the minus c.
So therefore, if we're dealing with a subtracted number,
that means we're going to be shifting to the right.
And if we see a positive c, that means
we're going to be shifting to the left.
Now, how far are we going to be shifted?
So here's where I usually like to start off with my students.
I say, all right.
Let's say let's do a general equation,
y equals sine of x minus pi.
And we get to this and you say, oh, well, now you're
going to shift pi units to the right.
So what we'll do is we'll take our whole trigonometric graph
and you're just going to shift it over pi units to the right
and then you graph it.
But then I always get students when
I start transforming like this.
And I say, cosine is going to equal, let's say, pi x plus pi.
And I say, well, that means now shifting to the left one unit.
And that's correct from what we've learned so far.
But it's important for us because now when we're talking,
we're not just talking about taking a function
and shifting it left or right.
But remember, our trigonometric functions are cyclical.
They keep on repeating on and on.
And there's a very important part
that's a part of the trigonometric functions, which
is the period.
And what that says is the distance,
how long it takes for a function to go and complete one
revolution or to repeat itself.
And that period affects our phase shift.
And remember, to determine the period,
we had to look at our b.
And so when we have a value of b,
that's going to affect how we're going to shift our graph left
or right.
So for this function, let's go back to our original function.
And on our original function, we said,
it's going to be 2 pi divided by b-- so 2 pi divided by 1,
which equals 2 pi.
However, for the cosine function,
when we look at our period, we're
going to take 2 pi divided by pi, which means now we're
going to have a period of 2.
So rather than it taking a distance of 2 pi,
it now takes a distance of 2 for this graph to repeat itself.
So how are we going to determine what the phase
shift is, how much the graph is going to shift left or right?
So what we do is we do the initial period.
And you're going to see me do this on all of my videos.
I always like to graph the initial period.
And I know and I don't want to be confusing with this
because graphs are cyclical.
They repeat on and on in the positive and negative
direction.
But when we look at the parent graph,
we look at the initial period, meaning
the first period we can graph.
And once we have that first graph, we can repeat.
So by looking at the initial period,
we need to understand where does the sine and cosine graph
start and end.
And both the graphs start at 0 and they also end at 2 pi.
Now, that can change, though, depending on the period
and on our phase shift.
So we could say that each graph always starts at 0
and always ends at 2 pi.
Now, if there is a transformation,
though, that's going to affect our initial period.
Now our initial period is going to start and end
at different points.
And what that's going to help us is
when we find our initial period, that's
going to tell us where our phase shift is.
So rather than our graph starting at 0,
if it starts at somewhere different,
now we know exactly what our phase shift is.
So let's do that for this graph.
If I wanted to find the phase shift, rather than just saying,
oh, it's pi to the right, let's do x minus pi equals 0.
Now when you add pi, what you notice
is you get x equals pi, which tells us
we shift our graph pi units to the right.
So when trying to find what is your phase shift,
take whatever's inside your function and set it equal to 0
because that's your initial starting
point of your initial period of your parent graph.
So let's go and take a look at this one.
Why is this not shifting left pi?
So I take my function, which is pi x plus pi,
and I set it equal to 0.
Now by solving for x, I subtract pi on both sides
and I get pi x equals negative pi.
Divide by pi and I get x equals negative 1.
So now you can see that when I graph this, rather than saying,
oh, remember the sine function-- we start at 0 then it goes up.
Or the cosine graph-- we say, oh, it starts at 0
and goes up to negative 1.
Now, that graph still is going to start at the y-value of 1
but now the x-value's going to be negative 1.
So what that means is now I'm shifting.
This graph has a phase shift of one to the left.
So there you go, ladies and gentlemen.
That is a definition of "phase shift"
for the sine and cosine graphs.
Thanks.