Introduction

In previous topics, we looked at how equations can implicitly define curves and how conics are structured. Now, we will learn how to represent these curves parametrically.

In Unit 4.1, we learned that parametric equations allow us to express both $x$ and $y$ as functions of a third variable, usually $t$ (time) or $\theta$ (angle), which gives us the ability to describe not just the shape of a curve, but also the direction and speed at which it is traced. Now, let’s connect this to the implicitly defined functions we covered in Unit 4.5 and 4.6.

Parametrizing Implicitly Defined Functions

An implicitly defined function is an equation involving $x$ and $y$ where $y$ is not necessarily isolated. A parametrization for such a function consists of two equations:

For these to be a valid parametrization, substituting $x(t)$ and $y(t)$ back into the original $x-y$ equation must satisfy that equation for every value of $t$ in the given domain.

Parametrizing Standard Functions

If you have a function where $y$ is already isolated ($y=f(x)$), parametrizing it is straightforward. We simply let $x$ be the “driver”:

Similarly, if an equation is solved for $x$ ($x=f(y)$), we let $y$ be the “driver”:

Figure 1: A diagram showing a graph of $f(x)=x^2$ and its parametrization, made using Desmos.

Inverses and Orientation

If a function $f$ is invertible, we can parametrize its inverse by simply swapping the roles of $t$:

This swapping of $x(t)$ and $y(t)$ effectively reflects the curve over the line $y=x$, which is the geometric definition of an inverse.

Parametrizing Conic Sections

While we can parametrize parabolas using the “driver” method from above, circles, ellipses, and hyperbolas require trigonometric functions to account for their closed loops or asymptotic behavior

A. Parabolas (Linear-Quadratic)

 Since a parabola is essentially a quadratic function of either $x$ or $y$, we use substitution:

B. Ellipses and Circles (Sine and Cosine)

An ellipse with center $(h,k)$, horizontal radius $a$, and vertical radius $b$ is defined by:

To satisfy this equation using trigonometry (remember that ${\cos }^{2}t+{\sin }^{2}t=1$!), we use:

Since the sine and cosine functions complete one full cycle as $t$ varies from $0$ to $2\pi$, choosing $0\le t\le 2\pi$ as our interval traces the entire ellipse (or circle) exactly once.

Figure 2: An ellipse with a center $(h,k)$ with a labeled horizontal and vertical radius, made using Desmos. 

Special Case: Circles

A circle is simply a type of ellipse whose horizontal and vertical radii are equal. If we let $a=b=r$, where $r$ is the radius of the circle, then the ellipse equation becomes $\frac{(x-h{)}^2}{r^2}+\frac{(y-k{)}^2}{r^2}=1$, which is the standard equation of a circle centered at $(h,k)$ with radius $r$. Substituting $a=b=r$ into the ellipse parametrization gives $x(t)=h+r\cos (t)$, $y(t)=k+r\sin (t)$. Since both coordinates are scaled by the same distance $r$, every point generated by the parametrization is exactly $r$ units from the center, producing a circle instead of an ellipse.

C. Hyperbolas (Secant and Tangent)

Hyperbolas use the identity ${\sec }^{2}t-{\tan }^{2}t=1$. The variable associated with the positive term in the hyperbola equation gets the $\sec t$ function.

Figure 3: A horizontal hyperbola, made using Desmos. Note that the positive term of the hyperbola contains the $\sec (t)$ expression.

Figure 4: A vertical hyperbola, made using Desmos.

$\begin{array}{ccc}\text{Conic Section}& \text{Standard Equation}& \text{Parametric Equations}\\ & & \\ \text{Parabola (Vertical)}& y-k=a(x-h{)}^2& x(t)=t\\ & & y(t)=k+a(t-h{)}^2\\ & & \\ & & \\ \text{Parabola (Horizontal)}& x-h=a(y-k{)}^2& x(t)=h+a(t-k{)}^2\\ & & y(t)=t\\ & & \\ & & \\ \text{Ellipse}& \frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1& x(t)=h+a\cos t\\ & & y(t)=k+b\sin t\\ & & \\ & & \\ \text{Hyperbola (Horizontal)}& \frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1& x(t)=h+a\sec t\\ & & y(t)=k+b\tan t\\ & & \\ & & \\ \text{Hyperbola (Vertical)}& \frac{(y-k{)}^2}{b^2}-\frac{(x-h{)}^2}{a^2}=1& x(t)=h+a\tan t\\ & & y(t)=k+b\sec t\\ & & \end{array}$  

Practice Problems