Parametric Equations

Parametric Equations

A system of equations with more than one dependent variable. Often parametric equations are used to represent the position of a moving point.

Two parametric equation examples: x=4t−t², y=2^t (moving point on x-y plane); x=s+t+1, y=2s−t, z=4s+7 (plane in space).

Key Formula

x = f(t), \quad y = g(t)

Where:

$x$ = The x-coordinate, expressed as a function of the parameter t
$y$ = The y-coordinate, expressed as a function of the parameter t
$t$ = The parameter (independent variable), often representing time
$f(t)$ = A function that gives the x-value for each value of t
$g(t)$ = A function that gives the y-value for each value of t

Worked Example

Problem: A point moves along a path defined by x = 2t and y = t² − 1. Find the (x, y) coordinates for t = 0, 1, 2, and 3, then eliminate the parameter to write y as a function of x.

Step 1: Substitute each value of t into both equations to find the coordinates.

t = 0: \; x = 0, \; y = -1 \quad \Rightarrow (0, -1)

Step 2: Continue for the remaining values of t.

t = 1: (2, 0), \quad t = 2: (4, 3), \quad t = 3: (6, 8)

Step 3: To eliminate the parameter, solve the x-equation for t.

x = 2t \;\Rightarrow\; t = \frac{x}{2}

Step 4: Substitute this expression for t into the y-equation.

y = \left(\frac{x}{2}\right)^2 - 1 = \frac{x^2}{4} - 1

Answer: The rectangular (Cartesian) equation is y = x²/4 − 1, a parabola. The parametric form also tells you the direction of travel: as t increases from 0 to 3, the point moves from (0, −1) to (6, 8).

Another Example

This example shows a closed curve (a circle) that cannot be written as a single function y = f(x), demonstrating a key advantage of parametric equations over standard rectangular form.

Problem: Write parametric equations for a circle of radius 3 centered at the origin, then verify that the point at t = π/4 lies on the circle x² + y² = 9.

Step 1: The standard parametric form of a circle of radius r centered at the origin uses cosine and sine.

x = 3\cos t, \quad y = 3\sin t, \quad 0 \le t < 2\pi

Step 2: Evaluate both equations at t = π/4.

x = 3\cos\frac{\pi}{4} = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}, \quad y = 3\sin\frac{\pi}{4} = \frac{3\sqrt{2}}{2}

Step 3: Check that x² + y² = 9.

\left(\frac{3\sqrt{2}}{2}\right)^2 + \left(\frac{3\sqrt{2}}{2}\right)^2 = \frac{18}{4} + \frac{18}{4} = \frac{36}{4} = 9 \; \checkmark

Answer: The parametric equations x = 3cos t, y = 3sin t correctly trace a circle of radius 3. At t = π/4 the point is (3√2/2, 3√2/2), which satisfies x² + y² = 9.

Frequently Asked Questions

How do you convert parametric equations to a rectangular (Cartesian) equation?

Solve one parametric equation for the parameter t, then substitute that expression into the other equation to eliminate t. For instance, if x = 2t and y = t + 5, solve the first equation for t = x/2 and substitute into the second to get y = x/2 + 5. Sometimes you may need a trigonometric identity (like sin²t + cos²t = 1) instead of direct substitution.

What is the difference between parametric equations and a regular equation?

A regular (rectangular) equation relates x and y directly, such as y = x². Parametric equations introduce a third variable, the parameter t, and express x and y separately as functions of t. This lets you describe curves that fail the vertical line test (like circles) and also encodes direction and speed of motion along the curve.

When do you use parametric equations?

You use parametric equations when you need to describe motion along a path (tracking position over time), when a curve cannot be expressed as a single function y = f(x), or when you want to model projectile trajectories, circular motion, and animations. They are fundamental in physics, computer graphics, and multivariable calculus.

Parametric Equations vs. Rectangular (Cartesian) Equation

	Parametric Equations	Rectangular (Cartesian) Equation
Form	x = f(t), y = g(t) — two equations with a parameter	A single equation relating x and y directly, e.g. y = x² + 1
Direction of travel	Built in — increasing t traces the curve in a specific direction	Not included — the equation describes the shape only
Curves like circles	Easily represented, e.g. x = r cos t, y = r sin t	Requires splitting into upper and lower halves or using an implicit equation
Number of variables	Three: x, y, and the parameter t	Two: x and y
Typical use	Motion, animation, physics trajectories	Graphing static curves, algebra

Why It Matters

Parametric equations appear throughout precalculus, AP Calculus BC, and physics courses. In physics, projectile motion is naturally modeled with parametric equations where x and y positions depend on time. In computer graphics and engineering, parametric curves (including Bézier curves) are the standard way to define paths, animations, and shapes.

Common Mistakes

Mistake: Forgetting to restrict the domain of the rectangular equation after eliminating the parameter.

Correction: The parameter t often has a limited range that restricts which part of the curve is traced. For example, if t ≥ 0 and x = 2t, then x ≥ 0, so the rectangular equation only applies for x ≥ 0. Always check whether the original parameter range limits x or y.

Mistake: Assuming that every set of parametric equations traces the entire curve of the corresponding rectangular equation.

Correction: Different parametrizations of the same rectangular curve can trace different portions or directions. For instance, x = t² and y = t⁴ gives y = x² but only for x ≥ 0, while x = t and y = t² traces the full parabola.

Related Terms

Parameter (Algebra) — The independent variable (usually t) in parametric equations
Parametrize — The process of writing a curve in parametric form
Dependent Variable — x and y are dependent variables in parametric equations
Parametric Derivative Formulas — Finding dy/dx from parametric equations using dy/dt ÷ dx/dt
Area Using Parametric Equations — Computing enclosed area from parametric curves
Polar Coordinates — Another alternative to Cartesian form for describing curves
Simultaneous Equations — Parametric equations are a system solved simultaneously
Point — Each value of t produces a point (x, y) on the curve