Welcome to
Introduction to Parametric Equations.
The goals of this video are to define a parametric equation
and also to graph a parametric equation.
So up until now we've always had our plane curves
described implicitly in terms of x,
meaning it was solved for y,
and all the terms involving x or the constants
were on the other side of the equal sign.
Or sometimes we had an equation
that was implicitly in terms of y,
where it was solved for x and all the y terms
or constants were on the other side of the equals sign.
Well, sometimes it's useful to have
a third variable or a parameter.
And for parametric equations this parameter is usually t.
So when both x and y are defined as a function of t,
we have parametric equations.
So we have x equals some function in terms of t,
and y equals some function in terms of t.
A great example of the use of parametric equations
would be the motion of a projectile.
So if you can imagine being on top of
a roof that is 12 feet high,
and let's say you kick a soccer ball off the roof,
this yellow plane curve would model the path of the ball
until it hits the ground.
So as time passes,
we can see that ball moving through the air,
until it hits the ground.
And the benefit of using a parametric equation
to model this path, is
if we have x equal to some function in terms of time,
and y equal to some function in terms of time,
for any given value of t,
x would represent the horizontal distance traveled,
and y would represent the height of the object
at any point in time.
Before we take a look at some more plane curves,
let's talk a little bit more
about using parametric equations.
T is usually defined over a particular interval,
but not always.
The values of t control the values of x and y,
but are not used when plotting the points
on the coordinate plane.
However, t does trace the plane curve
in a particular direction as t increases.
And we can show the direction
by using arrows along the plane curve.
Before we do some of our own graphing,
let's go ahead and take a look at some more plane curves
generated by parametric equations.
What's neat about parametric equations
is we get some very interesting plane curves
that we haven't seen before.
Well, let's start by taking a look at a basic equation
where we have x equals two minus three t,
and y equals one minus two t.
Notice as t increases from zero to one,
the plane curve is traced in a particular direction,
as we see here.
So this parametric equation is just a line.
Now let's take a look at the graph of
x equals t squared, and y equals t.
Again, notice the particular direction
in which the plane curve is traced.
Here we have x equals cosine t, y equals sine t,
and we'll trace this curve from zero to two pi.
So in this case, t would represent the angle.
Hopefully this is not a surprise,
because this should remind us of the unit circle.
Let's go and take a look at two more.
Here's x equals sine three t and y equals sine four t.
So as you can see, we get some very interesting
and cool looking graphs from parametric equations.
Let's go and take a look at one more.
Here we have the graph of x equals cosine t,
and y equals sine of the quantity t plus sine five t.
Let's go ahead and go back and graph
some of our own parametric equations.
So if we want to graph the parametric equation by hand
where we have x equals two t minus one,
y equals negative t plus three,
and t is on the closed interval from negative four to five.
So we'll select values of t in this interval
and then determine the value of x and y
based upon the value of t.
So when t is equal to negative four,
we'd have x equals negative nine,
and y would be positive seven.
So we plot the point negative nine, seven
on the coordinate plane.
When t is negative two,
we'd have negative four minus one,
so x would be negative five,
and y would be negative negative two,
plus three, which is positive five.
So we'd have to plot the point negative five, five.
When t is zero, we'll have x equals negative one,
and y equals positive three.
When t is two, we'll have four minus one for x,
that'll be three, and y will be positive one.
We'll plot the point three, one.
And you can probably tell by now we do have a line,
let's go ahead and jump to t equals five,
when t is five, x will be nine,
and for y we'll have negative five plus three,
which is negative two.
So we'll have the point nine, negative two, which is here.
Notice it also asks us to indicate
the direction in which the plane curve is traced
as t increases.
We'll go ahead and graph this line segment,
but notice as t increase, we plot the points
in this direction,
therefore we'll indicate the direction
using arrows along this line segment.
Let's go and take a look at another one.
Here we have x equals four cosine t,
and y equals four sine t,
and t is in the closed interval from zero to two pi.
So since we have trig functions here,
I went ahead and selected the values for t
as zero, pi over two, pi, three pi over two, and two pi.
So when t equals zero we have four times cosine zero.
Well cosine zero was one, so x is four.
Well the sine of zero is zero so y will be zero,
so we have the point four, zero, should be here.
Next we have t equals pi over two.
Well cosine pi over two is zero, so x is zero.
The sine of pi over two would be one,
so we have four times one,
so now y is four, so we have the point zero, four.
Next we have t equals pi.
Well cosine pi is negative one,
so we'll have four times negative one or negative four,
and the sine of pi is equal to zero, so y is zero.
so we have a point negative four, zero, which is here.
Now we have t equals three pi over two,
and the cosine of three pi over two is zero,
so x is zero.
Sine of three pi over two is negative one,
so y would be negative four.
So now we're down here.
And then when t is pi over two, x is equal to four,
and y is equal to zero, so we're back where we started.
Now these equations may look familiar.
Remember on the unit circle,
x is equal to cosine theta,
and y is equal to sine theta.
So this plane curve is a circle,
where the center is at the origin,
and r is equal to four.
If we didn't recognize this,
we could select additional values of t
to find additional points, but the graph will be a circle.
And we did plot the points counterclockwise,
so we'll indicate that using these arrows,
showing how the circle would be traced,
as t increases from zero to two pi.
Okay, that's going to do it for this video.
In the next two videos we'll take a look
at how to rewrite parametric equations
in rectangular form, as well as how to use
our graphing calculators to graph parametric equations.
I hope you found this video helpful.