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## Algebra (all content)

# Parametric equations with the same graph

Sal shows how different parametric equations can result in the same relationship between x and y (and therefore with the same graph). Created by Sal Khan.

## Video transcript

In the last video, we started
with these parametric equations: x is equal to
3 cosine of t and y is equal to 2 sine of t. And doing a little bit of
algebra, we were able to remove the parameter and turn it into
an equation that we normally associate with an ellipse. We got x squared over 9 plus y
squared over 4 is equal to 1. And we graphed it. And I'll do that. Actually I have it from the
previous-- We graphed it and we got something that looked
like that right there. And we said well why
don't we always just represent it like this? Well this right here-- this
equation just in terms of x and y --it just
represents the shape. It doesn't represent the actual
path that-- Let's say that t represents an object's point--
an object's position at some place and time. And we went over that. And as time progressed,
we went around the ellipse, in that case. And as time goes forward, we
kept going around this ellipse. But this begs the question: is
this the only set of parametric equations that, when we do a
little bit of algebra, gives this path? Can we think of another set
that gives-- that would also, when you do the
algebra, get here? And I think we can. So let's try a couple. I mean, we could do a
whole bunch of them. Actually, maybe we'll do a
whole bunch of them, just to see that there-- And what
you'll see is that there's actually an infinite number
of parametric equations that have this path. So one could be x is
equal to 3 cosine of 2t. And the other one is y
is equal 2 sine of 2t. Like we did in the last video,
let's solve for cosine of 2t and solve for sine of 2t. So we just have to divide both
sides of-- Well, both sides of this top equation by 3. You get x over 3 is
equal to cosine of 2t. Divide both sides of
this equation by 2. You get y over 2 is
equal to sine of 2t. And we'll use the same trig
identities we did before. When we did this problem, we
said that sine squared of t plus cosine squared
of t is equal to 1. Well if that's true,
so is this true. Let me see. If that's true, then we can
also say that sine squared of 2t plus cosine squared
of 2t is equal to 1. In fact, sine squared of
anything here plus cosine squared of anything, as long as
it's the same anything as we put here-- I can't have a 2t
here and a 3t here, but if this is a 2 and this is a 2 --then
that's always going to be equal to 1. You can just fill in the black. Because this is always going
to be the same angle. So if this is true, we can then
just take this and substitute it in for cosine of 2t and take
the y over 2 and substitute it in for sine of 2t. And we get the sine of 2t
squared becomes y over 2 squared plus cosine of
2t is x over 3 squared. It equals 1. And that is exactly what
we had to begin with. That is the same thing as x
squared over 9-- I'm just switching the order here --plus
y squared over 4 is equal to 1. So both of these parametric
equations, both sets of parametric equations have
the exact same path. So how are they different? Let's see. If we were to take-- So in
the previous video, we took this yellow one. Actually I rewrote it here. And we did a little table
and we plotted the points that it hits as you
go around the circle. So let's do the same
thing for our new set. So just since I lost it, let me
write the two equations here. It was x is equal to 3
cosine of 2t and y is equal to 2 sine of 2t. Let me make a little table. There we go. And we're going to have a t. And we're going to
have an x and a y. And let's pick the
same values of t. 0 pi over 2 and pi. When t is 0, cosine
of 2 times 0. That's cosine of 0, which
is one, times 3, it's 3. When t is equal to pi over
2, 2 times pi over 2 is pi. Cosine of pi over 2 is minus 1. Minus 1 times 3 is minus 3. When t is equal to pi, cosine
of 2 times pi is 2 pi. Cosine of 2 pi is the same
thing as cosine of 0. So that's 1. 1 times 3 is 3. And then on the y side. When t is equal to 0,
that's the same thing as sine of 2 times 0. Sine of 0 times 2 is 0. Pi over 2. Sine of 2 times pi over 2. That's the same
thing as sine of 4. That's also 0. 0 times 2 is 0. And then pi. Sine of 2 pi. Well, that's also 0. So here we get a bunch of 0's. So what happened here? Let me do it in a
different color. So in our first situation-- So
they both-- We established that both this set of parametric
equations and this set of parametric equations, they both
have the same overall, I guess we could say, shape
of their path. They're both this ellipse. But what we saw in the previous
videos, when you plot these points at time t equals
0, we were right. This point right here
is that right there. And then at t is equal to pi
over 2, we got right there. And then at t is equal to
pi, we got over here. And we did that to establish
that the direction of motion. If these parametric equations
really are describing some type of motion, it would
be counterclockwise. And what happens in this
scenario-- So when t is equal to 0, we're still
at the point 3, 0. So they at least-- At t equals
0, you can kind of view them as starting at the same
point, if we assume we're starting a t equals 0. You don't have to
start at t equals 0. I did that a little bit in the
last video, but you can start at t equals minus a google
or t equals minus infinity. So you don't necessarily
have a start point. But if we assume t equals 0 as
a start point, you could say that they both started there. Then at-- let me pick a new
color that I haven't used yet --then at t is equal to
pi over 2-- Where are we? We're at minus 3, 0. We get all the way-- Oh. That's the same color
I used before. Let me see this. Let me do this color. We're here. Now notice: in the first one,
when we went from t equals 0 to t equals pi over 2, we
went from here to there. We went kind of a quarter of
the way around the ellipse. But now when we went from
t equals zero to pi over 2, where did we go? we? Went halfway around
the ellipse. We went all the way from there,
all the way over there. And likewise, when we went from
t equals pi over 2 to t equals pi with this set of parametric
equations, we went another quarter of the ellipse. We went from there to there. But here, when we go from t
equals pi ever 2 to t equals pi, we go all of this way. We go back to the beginning
part of our ellipse. So what you see is that this
set of parametric equations has the exact same shape of
its path as this set of parametric equations. Except it's going around it
at twice as fast of a rate. For every time when t increases
by pi over 2 here, we go by-- we kind of go a quarter
way around the ellipse. But when t increases by pi
over 2 here, we go halfway around the ellipse. So the thing to realize-- and I
know I've touched on this before --is that even though
both of these sets of parametric equations, when you
do the algebra, they can kind of be converted
into this shape. You lose the information about
where our particle is as it's rotating around the ellipse or
how fast it's rotating around the ellipse. And that's why you need
these parametric equations. We can even set up a parametric
equation that goes in the other direction. Instead of having these-- and I
encourage you to play with that --but if you instead of this,
if you just put a minus sign right here. Cosine of minus t and
2 sine of minus t. Instead of going in that
direction, it would go in this direction. It would go in a
clockwise direction. So one thing that you've
probably been thinking from the beginning is OK, I was able to
go from my parametric equations to this equation of ellipse
in terms of just x and y. Can you go back the other way? Could you go from this to this? And, I think you might realize
now, that the answer is no. Because there's no way, just
with the information that you're given here, to know
that you should go to this parametric equation or this
parametric equation or any of an infinite number of
parametric equations. I mean anything of the form x
is equal to 3 cosine of really anything times t and y is equal
to 3 times cosine of-- As long as it's the same anything-- I
drew the two squiggly marks the same. --as long as these two things
are the same, then you will have-- they'll both
converge to this shape. So if you have the shape, you
don't know what parametric equation you can go back to. You could make up one,
but you don't know which one it'll go back to. And just to kind of hit the
point home of kind of making up a parametric equation. Sometimes that's asked of you. So let's just do a
very simple example. Let's start with a
normal function of x. So let's say that we have
the equation y is equal to x squared plus x. I just made it up. And let's say that you were
asked to turn this into a parametric equation. And this is often a very hard
thing for people because there is not one right answer. You can turn into an arbitrary
number of parametric equations. I just may do something
really crazy and arbitrary. I could say x is equal-- When I
have y defined explicitly in terms of x like this, I can
make x into anything in terms of t. I could say it's cosine of t
minus the natural log of t. I just made that up. That's just a random thing. But then if x is this, then y
is going to be equal to-- I just substitute this back in
--cosine of t minus ln of t squared plus cosine
of t minus ln of t. And we're done. We just converted this into a
parametric equation or a set of parametric equations. I could've also just
written x is equal to t. And then what would y equal? y would be equal to
t squared plus t. You might say well what's
the difference between this parametric equation and
this parametric equation? Well, they're both
going to have the same shape of their paths. It's going to be something
like a parabola. But the rate and the direction
with which they progress on those paths will be
very very different. It's actually a very
interesting thing to think about. There are some paths that you
could take where-- Let's say that the-- You could set
up a parametric equation. Let's say that the shape is--
You know everything we've done so far, we've always been going
in one direction, but you could have a-- There are scenarios--
And maybe if I have time, I'll make a video. And this isn't what-- Let's say
the shape of the path is some type of, I dunno, let's say
it's a circle of some kind. I'm just going to--
That's not this. I'm doing a completely
different example right now. I'm just kind of-- Sometimes,
with the ellipses we had paths that went counterclockwise and
then paths that went clockwise. You can also have paths that
kind of isolate in between, move around, move back and
forth along the thing. So there's all sorts of
parametric equations that you can define. And you can say if t
equals from this to this, you'll use this set of
parametric equations. If it's another set of
t's, use another one. So there's all sorts of crazy
things you can do to say what happens as you move
along the path. So the difference-- Not that
this is the path of that. This is actually
more of a parabola. But the difference between
this and this is how you move along the shape. Anyway, I hope you found
that vaguely useful.