Chapter 10
Parametric Equations and Polar Coordinates
Math 1B
§ 10.1 Curves Defined by Parametric Equations
Suppose that a particle moves along the curve C. It is impossible to
describe C by an equation of the form y = f (x) because C fails the
Vertical Line Test. But the x and y coordinates of the particle are
functions of time and we can write x = f (t) and y = g(t).
There are also a great many curves out there that we can’t even write down as a single equation in
terms of only x and y. So, to deal with some of these problems we introduce parametric equations.
Instead of defining y in terms of x (y = f (x)) or x in terms of y (x = h(y)) we define both x and y in
terms of a third variable called a parameter as follows,
x = f (t)
y = g(t)
This third variable is usually denoted by t (as we did here) but doesn’t have to be of course. Sometimes
we will restrict the values of t that we’ll use and at other times we won’t.
Each value of t defines a point (x, y) = (f (t), g(t)) that we can plot. As t varies, the point
(x, y) = (f (t), g(t)) varies and traces out a curve C, which we will call a parametric curve.
Example: Sketch the curve defined by the parametric equations
x = t2 + t
t
x
y = 2t − 1
y
Example: Sketch the curve defined by the parametric equations
x = t2 + t
y = 2t − 1
−1 ≤ t ≤ 1
Note: In general, the curve with parametric equations
(
)
(
x = f (t )
)
has initial point f ( a ) , g ( a ) and terminal point f ( b) , g ( b) .
y = g (t )
a≤t ≤b
It is now time to take a look at an easier method of sketching this parametric curve. This method uses
the fact that in many, but not all, cases we can actually eliminate the parameter from the parametric
equations and get a function involving only x and y. There will be two small problems with this
method, but it will be easy to address those problems. It is important to note however that we won’t
always be able to do this.
Just how we eliminate the parameter will depend upon the parametric equations that we’ve got. Let’s
see how to eliminate the parameter for the set of parametric equations that we’ve been working with.
Example: Eliminate the parameter from the set of parametric equations to find a Cartesian equation of
the curve.
x = t2 + t
y = 2t − 1
−1 ≤ t ≤ 1
Note: There are two small problems with this method. The first is direction of motion. The equation involving only x and y
will NOT give the direction of motion of the parametric curve. This is an easy problem to fix however. All we need to do is
plug in some values of t into the parametric equations and we can determine direction of motion from that. We will see the
second problem in the next example.
Example: Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve and
indicate with an arrow the direction in which the curve is traced as the parameter increases.
x = 5cost
y = 2sint
0 ≤ t ≤ 2π
Example: Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve and
indicate with an arrow the direction in which the curve is traced as the parameter increases.
x = 5cos ( 3t )
y = 2sin ( 3t )
0 ≤ t ≤ 2π
The previous two examples show that different sets of parametric equations can represent the same
curve. So we distinguish between a curve, which is a set of points, and a parametric curve, in which
the points are traced in a particular way.
Example: Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve and
indicate with an arrow the direction in which the curve is traced as the parameter increases.
x = et − 1
y = e2t
Example: Describe the motion of the particle with position (x, y) as t varies in the given interval.
x = 3cost
y = 1+ cos 2 t
0 ≤ t ≤ 2π