Sometimes you’ll want to convert parametric equations back into a single equation relating [latex]x[/latex] and [latex]y[/latex]. This process is called eliminating the parameter, and it helps you identify what type of curve you’re working with. The basic strategy is to solve one parametric equation for the parameter [latex]t[/latex], then substitute this expression into the other equation.
Given the parametric equations: [latex]x(t) = t^2 - 3, \quad y(t) = 2t + 1, \quad -2 \leq t \leq 3[/latex]
Step 1: Solve the simpler equation for [latex]t[/latex] from [latex]y = 2t + 1[/latex]:
Result: This is a parabola opening to the right with [latex]x[/latex] as a function of [latex]y[/latex].
When you eliminate the parameter, don’t forget about the original restrictions on [latex]t[/latex]. In the example above, [latex]-2 \leq t \leq 3[/latex] means the curve only exists between specific endpoints, not as an infinite parabola.
Before working through more examples, recall these essential relationships.
Key Identities
Pythagorean Identity: For any angle [latex]t[/latex], [latex]\sin^2 t + \cos^2 t = 1[/latex]
Circle Equation: A circle of radius [latex]a[/latex] centered at the origin has equation [latex]x^2 + y^2 = a^2[/latex]
These identities are particularly useful when your parametric equations involve trigonometric functions, which often describe circular or elliptical curves.
Eliminate the parameter for each of the plane curves described by the following parametric equations and describe the resulting graph.
Note that when we square both sides it is important to observe that [latex]x\ge 0[/latex]. Substituting [latex]t=\frac{{x}^{2}-4}{2}[/latex] this into [latex]y\left(t\right)[/latex] yields
This is the equation of a parabola opening upward. There is, however, a domain restriction because of the limits on the parameter t. When [latex]t=-2[/latex], [latex]x=\sqrt{2\left(-2\right)+4}=0[/latex], and when [latex]t=6[/latex], [latex]x=\sqrt{2\left(6\right)+4}=4[/latex]. The graph of this plane curve follows.
Figure 7. Graph of the plane curve described by the parametric equations in part a.
Sometimes it is necessary to be a bit creative in eliminating the parameter. The parametric equations for this example are
Solving either equation for t directly is not advisable because sine and cosine are not one-to-one functions. However, dividing the first equation by 4 and the second equation by 3 (and suppressing the t) gives us
Now use the Pythagorean identity [latex]{\cos}^{2}t+{\sin}^{2}t=1[/latex] and replace the expressions for [latex]\sin{t}[/latex] and [latex]\cos{t}[/latex] with the equivalent expressions in terms of x and y. This gives
This is the equation of a horizontal ellipse centered at the origin, with semimajor axis 4 and semiminor axis 3 as shown in the following graph.
Figure 8. Graph of the plane curve described by the parametric equations in part b.
As t progresses from [latex]0[/latex] to [latex]2\pi[/latex], a point on the curve traverses the ellipse once, in a counterclockwise direction. Recall from the section opener that the orbit of Earth around the Sun is also elliptical. This is a perfect example of using parameterized curves to model a real-world phenomenon.
Watch the following video to see the worked solution to the example above.
For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.
So far we have seen the method of eliminating the parameter, assuming we know a set of parametric equations that describe a plane curve. What if we would like to start with the equation of a curve and determine a pair of parametric equations for that curve? This is certainly possible, and in fact it is possible to do so in many different ways for a given curve. The process is known as parameterization of a curve.
Find two different pairs of parametric equations to represent the graph of [latex]y=2{x}^{2}-3[/latex].
First, it is always possible to parameterize a curve by defining [latex]x\left(t\right)=t[/latex], then replacing x with t in the equation for [latex]y\left(t\right)[/latex]. This gives the parameterization
Since there is no restriction on the domain in the original graph, there is no restriction on the values of t.
We have complete freedom in the choice for the second parameterization. For example, we can choose [latex]x\left(t\right)=3t - 2[/latex]. The only thing we need to check is that there are no restrictions imposed on x; that is, the range of [latex]x\left(t\right)[/latex] is all real numbers. This is the case for [latex]x\left(t\right)=3t - 2[/latex]. Now since [latex]y=2{x}^{2}-3[/latex], we can substitute [latex]x\left(t\right)=3t - 2[/latex] for x. This gives
Watch the following video to see the worked solution to the example above.
For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.
