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Parametrize — Definition, Meaning & Examples

Parametrize

To write in terms of parametric equations.

 

Example:

The line x + y = 2 can be parametrized as x = 1 + t, y = 1 – t.

 

 

See also

Curve

Worked Example

Problem: Parametrize the circle x² + y² = 9.
Step 1: Recognize that any point on a circle of radius r can be written using sine and cosine. Here r = 3.
Step 2: Let t be the parameter representing the angle measured from the positive x-axis. Write x and y in terms of t.
x=3cos⁡t,y=3sin⁡t,0≤t<2πx = 3\cos t, \quad y = 3\sin t, \quad 0 \le t < 2\pi
Step 3: Verify: substitute back into the original equation.
(3cos⁡t)2+(3sin⁡t)2=9cos⁡2t+9sin⁡2t=9(3\cos t)^2 + (3\sin t)^2 = 9\cos^2 t + 9\sin^2 t = 9
Answer: A valid parametrization of the circle is x = 3cos t, y = 3sin t for 0 ≤ t < 2π.

Why It Matters

Parametrization lets you trace a path along a curve with a single variable, which is essential for computing arc length, line integrals, and animations. It also allows you to describe curves that cannot be written as a single function y = f(x), such as circles or figure-eights.

Common Mistakes

Mistake: Thinking there is only one correct parametrization for a given curve.
Correction: Most curves have infinitely many valid parametrizations. For example, x = 2 + t, y = −t and x = t, y = 2 − t both parametrize the line x + y = 2. Different parametrizations may trace the curve at different speeds or in different directions.

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