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MathidiaMathematics Academy
Parametric equations
Lesson 21 of 75
Introduction

Parametric representations

Objective. Generate coordinates and eliminate parameters.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Parametric representations define x and y separately in terms of a third variable t, called the parameter, rather than as a direct relation between x and y. Example: x=t², y=2t traces a parabola, which can be converted to Cartesian form by eliminating t: from y=2t, t=y/2, so x=(y/2)²=y²/4. Eliminating the parameter is useful when a standard Cartesian equation is needed for graphing or comparison.

Worked example

A curve has parametric equations x=3t+1, y=t²−2. Find the Cartesian equation by eliminating the parameter.

3 … 1
Coordinate grid with the line 3 … 1
  1. 1Rearrange x=3t+1 to make t the subject: t=(x−1)/3
  2. 2Substitute into y=t²−2
  3. 3Write y=((x−1)/3)²−2
  4. 4Simplify the squared term: y=(x−1)²/9−2

Answer: y=(x−1)²/9−2

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Common mistake. Students substitute t values inconsistently between the x and y equations, generating coordinate pairs that do not correspond to the same t value; always use the same t for both x and y simultaneously.

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