Professor Dave here, let’s talk about parametric equations.
By now we are quite familiar with equations where Y is expressed in terms of X, and we
have learned about a variety of different types of equations that work this way.
We also just learned about polar equations, where R is expressed in terms of theta.
But there is one more type of equation we will want to briefly mention, and those are
parametric equations.
These are sets of equations in which the X and Y values are calculated in separate equations,
and each dependent on some other variable.
The most common application of this involves two-dimensional motion of objects, like projectile
motion.
We learned all about this in the classical physics course, so you can head there if you
want more information about this subject, but the basic gist of it is that when you
throw a ball into the air at an angle, its X and Y motion are independent of one another.
It’s horizontal velocity will be constant, if we ignore wind resistance, but its vertical
velocity will be constantly decreasing from the moment of release, because gravity will
be continuously accelerating the ball in the negative direction, towards the ground.
So in such a case, it can make sense to have an equation that models the X coordinate of
the ball with respect to time, and an equation the models the Y coordinate of the ball with
respect to time, as they do indeed depend on the parameter of time differently, and
we may sometimes want to ask a question about one of these values and not the other.
So we can say that in general, sets of parametric equations come in the form of X equals F of
T, and Y equals G of T. These are two separate functions, but they depend on the same parameter.
Then to graph the plane curve that results, we can just plug in some values of T and see
what we get.
For example, let’s look at the parametric equations X equals 2T and Y equals T squared.
Let’s make a little table, pick some T values, and then find the corresponding X and Y values.
Once we have all of these, we essentially have a set of X and Y coordinates, so we can
now plot all of these points, connect them, and we see that we get a parabola.
We can quite easily get the equation for this parabola by doing something called eliminating
the variable.
This is where we solve for T in one of the equations and plug it into the other.
Using this one, T must be equal to X over two, so let’s plug X over two into the other
equation, and we get Y equals X squared over four, or one fourth X squared.
Depending on the situation, it can be easier to plot points using the parametric equations,
or to find a way to eliminate the parameter and just graph it normally.
There are different situations that would favor one approach over the other, for example,
if using a graphing calculator, it can be easier to enter the equation of something
like an ellipse if parametric form rather than rectangular.
But an ability to understand how to move between these two types of equations is all we really
need, so let’s check comprehension.