Instead of relating y to x directly, both are given as functions of a parameter t. Each value of t names one point.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Usually time
When t is time, the equations describe not just the path but when the object reaches each point.
What it adds
A rectangular equation gives the path. Parametric equations give the path, the direction and the speed.
Eliminating the parameter
Solve one equation for t and substitute into the other. What remains is the rectangular equation of the path.
What elimination loses
The rectangular form forgets direction and any restriction on t, so a half circle can become a whole one.
Orientation
Orientation is the direction of travel as t increases, drawn as an arrow. It is the feature only the parametric form carries.
A third variable drives both coordinates
x and y are each given as functions of a parameter t. As t runs, the point traces a curve. The parameter is often time, and the form describes motion rather than just shape.
It describes what functions cannot
A circle fails the vertical line test and has no representation as y = f(x). Parametrically it is x = cos t, y = sin t with no difficulty. Parametric form escapes the single-output restriction.
Direction is part of the answer
Two parametrisations can trace the same path in opposite directions or at different speeds. Sketches should carry arrows, and descriptions should state the direction — the curve alone does not capture it.
Eliminating the parameter loses information
Solving for t and substituting recovers a Cartesian equation that identifies the shape but discards timing, direction and the interval. The conversion is genuinely lossy, which is worth stating explicitly.
Step 2: Try It Yourself
Tap and try it out.
- Angle45° = π/4 rad
- x-coordinate0.707
- y-coordinate0.707
- cos 45°0.707
- sin 45°0.707
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Kofi Eliminate a Parameter
Kofi has x = t + 1 and y = 2t.
- Step 1
He solves the first equation for t, giving t = x − 1.
Step 4: Your Turn
Practice makes it stick.
The Position
Problem 1 of 2
x = 3t and y = t + 2. What is x when t = 4?
The Circle
Problem 2 of 2
x = cos t and y = sin t. What is x² + y²?
Follow the Parameter
1 of 8
x = 2t, y = t². What is x when t = 3?
2 of 8
Same equations. What is y when t = 3?
3 of 8
x = t + 5, y = t. What is y when x = 9?
4 of 8
x = 4t, y = 8t. Eliminating t gives y = kx. What is k?
5 of 8
x = −t, y = 3t. As t increases, does x rise or fall? 1 rise, 2 fall.
6 of 8
x = t, y = t². Eliminating t gives y = xⁿ. What is n?
7 of 8
What does a parametric form carry that a rectangular one does not?
8 of 8
x = 5t, y = 3t. What is y when t = 2?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x = t − 1, y = 3t. What is y when t = 5?
Question 2 of 2
What is lost when the parameter is eliminated?
What You Learned
- Parametric functions give x and y as functions of a parameter t.
- Eliminating t recovers the rectangular equation of the path.
- Only the parametric form carries direction and timing.