Parametric equations, explained with examples

By The STEM-Tutor team · Updated October 2026

Parametric equations describe a curve by giving x and y separately in terms of a third variable, the parameter. They are in the second year of A level Maths and link algebra, trigonometry and calculus.

What a parameter is

Instead of one equation linking x and y, a parametric curve uses two: x = f(t) and y = g(t). The variable t is the parameter. Each value of t gives a point (x, y), and as t changes, the point traces out the curve. A helpful picture is a moving object: t is time, and the two equations tell you where the object is at each moment. For example, x = t, y = t² traces the parabola y = x², and x = cos t, y = sin t traces the unit circle.

Parametric equations can describe curves that are awkward or impossible to write as y = f(x), such as circles, loops and paths that cross themselves. The domain of t, such as 0 ≤ t ≤ 2π, decides which part of the curve is drawn.

Sketching from a table

To sketch a parametric curve, choose values of t, calculate x and y for each, and plot the points in order. Note where the curve meets the axes by setting x = 0 or y = 0 and solving for t.

Example 1: points and intercepts

A curve has x = t + 1, y = t² − 4 for −3 ≤ t ≤ 3. Find where it meets the axes.

• y = 0 when t = ±2, giving x = 3 or x = −1: points (3, 0) and (−1, 0).
• x = 0 when t = −1, giving y = −3: point (0, −3).

Converting to Cartesian form

To find the Cartesian equation, eliminate t. There are two main methods.

• Rearrange and substitute: make t the subject of one equation and substitute into the other.
• Use an identity: when the equations involve trig functions, use sin²t + cos²t = 1, 1 + tan²t = sec²t, or a double angle formula.

Example 2: substitution

Find the Cartesian equation of x = t + 1, y = t² − 4.

• t = x − 1, so y = (x − 1)² − 4.
• y = x² − 2x − 3. With −3 ≤ t ≤ 3, the domain is −2 ≤ x ≤ 4.

Example 3: a circle

Find the Cartesian equation of x = 3 + 2cos t, y = −1 + 2sin t.

• cos t = (x − 3)/2 and sin t = (y + 1)/2. Use cos²t + sin²t = 1.
• (x − 3)² + (y + 1)² = 4: a circle with centre (3, −1) and radius 2.

Example 4: a double angle

Find the Cartesian equation of x = sin t, y = cos 2t.

• cos 2t = 1 − 2sin²t.
• y = 1 − 2x², for −1 ≤ x ≤ 1, because x = sin t cannot go outside that range.

Differentiating parametric equations

The chain rule gives the gradient directly from the parametric form:

dy/dx = (dy/dt) / (dx/dt)

This gives dy/dx in terms of t. To find a tangent or normal at a point, find the value of t at that point first. A horizontal tangent occurs where dy/dt = 0, and a vertical tangent where dx/dt = 0.

Example 5: a tangent

Find the equation of the tangent to x = t², y = t³ at t = 2.

• The point is (4, 8). dx/dt = 2t and dy/dt = 3t², so dy/dx = 3t/2.
• At t = 2, dy/dx = 3. Tangent: y − 8 = 3(x − 4), so y = 3x − 4.

Example 6: stationary points

Find the stationary points of x = t², y = t³ − 3t.

• dy/dt = 3t² − 3 = 0 gives t = ±1 (dx/dt = 2t is not zero there).
• t = 1: (1, −2). t = −1: (1, 2). Two different points share an x-coordinate, so this curve is not a function y = f(x).

Example 7: a normal

Find the normal to x = 2cos t, y = 3sin t at t = π/4.

• Point: (√2, 3√2/2). dy/dx = 3cos t/(−2sin t) = −(3/2)cot t = −3/2 at t = π/4.
• Normal gradient 2/3: y − 3√2/2 = (2/3)(x − √2).

Areas under parametric curves

The area under a curve is ∫ y dx. With parametric equations, change the variable to t:

Area = ∫ y (dx/dt) dt, with the limits changed to the t-values at the ends.

Example 8: an area

Find the area between x = t², y = 2t, the x-axis and the line x = 4, for t ≥ 0.

• x = 0 at t = 0 and x = 4 at t = 2. dx/dt = 2t.
• Area = ∫₀² 2t · 2t dt = ∫₀² 4t² dt = [4t³/3]₀² = 32/3.
• Check: the Cartesian form is y = 2√x, and ∫₀⁴ 2x^1/2 dx = (4/3)(8) = 32/3. It matches.

Parametric equations in modelling

Projectile motion is naturally parametric. A ball launched at speed U and angle α has x = Ut cosα and y = Ut sinα − ½gt², with t as time. Questions may ask for the time of flight (set y = 0), the range (find x at that time), or the Cartesian path (eliminate t to get a parabola). Always state what t represents and give a sensible domain, such as t ≥ 0 until the ball lands.

Common mistakes

• Writing dy/dx = (dx/dt)/(dy/dt), upside down.
• Forgetting to change the limits to t-values when finding an area.
• Ignoring the domain of t, which restricts the Cartesian curve.
• Leaving the tangent equation in terms of t instead of finding the point first.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are parametric equations in AS or A level Maths?

They are in the second year of A level Maths for AQA, Edexcel and OCR.

Are parametric equations in AP Calculus?

Yes, but only in AP Calculus BC, in Unit 9, along with polar coordinates and vector-valued functions.

Key takeaways

• x and y are both given in terms of a parameter t.
• Eliminate t by substitution or with a trig identity.
• dy/dx = (dy/dt)/(dx/dt).
• Area = ∫ y (dx/dt) dt with t-limits.
• Keep track of the domain of t.

Related guides

• Trigonometric identities
• The chain rule
• A level Maths differentiation