Instead of relating y to x directly, give both as functions of a third variable 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 where the curve goes but when the object is at each place.
What it adds
A rectangular equation gives the path. Parametric equations give the path, the direction of travel and the speed.
Eliminating the parameter
Solve one equation for t and substitute into the other. What remains is the ordinary rectangular equation of the path.
The trigonometric trick
For x = cos t and y = sin t, squaring and adding gives x² + y² = 1. The Pythagorean identity removes t in one move.
What elimination loses
The rectangular form forgets direction and any restriction on t, so a half-circle can turn into a whole one.
A third variable drives both coordinates
Instead of y as a function of x, both x and y are given as functions of a parameter t. As t runs, the point (x(t), y(t)) traces a curve. The parameter is often time, but it need not be.
What it lets you describe
A circle fails the vertical line test and cannot be written as one function of x. Parametrically it is x = cos t, y = sin t, with no difficulty at all. Parametric form describes curves that functions cannot.
Eliminating the parameter
Solving one equation for t and substituting into the other recovers a Cartesian equation. It identifies the shape but discards the timing and direction, so information is genuinely lost in the conversion.
The parameter carries information the curve does not
Two parametrisations can trace the same path at different speeds, in different directions, or over different intervals. That extra information is exactly what makes the form useful in physics and animation.
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 Elena Eliminate the Parameter
Elena has x = t + 1 and y = 2t.
- Step 1
She 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 Other Coordinate
Problem 2 of 2
Same equations. What is y when t = 4?
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 = cos t, y = sin t. What is x² + y²?
5 of 8
x = 4t, y = 8t. Eliminating t gives y = kx. What is k?
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 equations give x and y as functions of a parameter t.
- Eliminating t recovers the rectangular equation of the path.
- The parametric form also carries direction and timing, which the rectangular form forgets.