Parametric Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parametric Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A way of defining a curve by expressing both x and y as separate functions of a third variable (parameter), typically t: x = f(t), y = g(t).

Instead of saying 'y depends on x,' parametric equations say 'both x and y depend on time t.' Imagine an ant walking on a table—at each moment t, the ant has an x-position and a y-position. The path it traces is the parametric curve, and t is the clock ticking forward.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Parametric equations decouple x and y, giving each its own equation in terms of a parameter. This allows curves that fail the vertical line test (like circles and loops) and naturally encodes direction and speed of motion along the curve.

Common stuck point: The same curve can have many different parametrizations. x = \cos t, y = \sin t and x = \cos(2t), y = \sin(2t) trace the same circle but at different speeds.

Sense of Study hint: Make a three-column table: t, x(t), y(t). Compute values for several t, then plot the (x, y) points and connect them in order of increasing t.

Worked Examples

Example 1

easy
Eliminate the parameter from x = 2t + 1 and y = t - 3 to find the rectangular equation.

Solution

  1. 1
    Solve the x-equation for t: t = \frac{x - 1}{2}.
  2. 2
    Substitute into the y-equation: y = \frac{x - 1}{2} - 3.
  3. 3
    Simplify: y = \frac{x - 1}{2} - 3 = \frac{x - 1 - 6}{2} = \frac{x - 7}{2}.

Answer

y = \frac{x - 7}{2}
Eliminating the parameter converts parametric equations back to a single rectangular equation. Solve one equation for t and substitute into the other. The result here is a line with slope \frac{1}{2} and y-intercept -\frac{7}{2}.

Example 2

medium
Eliminate the parameter from x = 3\cos(t) and y = 3\sin(t).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find parametric equations for the line through (1, 4) and (5, -2).

Example 2

hard
Eliminate the parameter from x = t^2 - 1 and y = t^3 - t and describe the curve.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitiontrigonometric functions