8.1PureCore

Parametric Equations

Instead of one equation linking x and y, parametric equations give both coordinates in terms of a third variable, the parameter t. This is natural for motion and for curves that fail the vertical line test.

26 min Parametric Equations

Edexcel A level Maths: 8.1 Parametric Equations (Domain and Range)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Understand x = f(t), y = g(t)
Convert parametric to Cartesian by eliminating t
Find the domain and range of a parametric curve
Plot points from parameter values

The idea

A parametric curve is given by $x = f(t)$ and $y = g(t)$ . Each value of $t$ produces a point $(x, y)$ . Eliminating $t$ recovers the Cartesian equation.

x = f(t), \quad y = g(t)

Both coordinates depend on the parameter t.

Eliminating the parameter

Make $t$ the subject of one equation and substitute into the other. The domain of $x$ becomes the domain of the Cartesian curve; the range of $y$ becomes its range.

Example 1Convert

x = t + 1

y = t^2

to Cartesian form.

1From the first,

t = x - 1

2Substitute:

y = (x-1)^2

Answer

y = (x-1)^2

Tip — The parameter’s range controls the domain and range of the final curve.

Formula recap

x = f(t), \ y = g(t)

Parametric form.

t = f^{-1}(x) \Rightarrow y = g(f^{-1}(x))

Eliminate t.

Common mistakes to avoid

Ignoring the range of t.

The parameter range restricts the domain/range of the Cartesian curve.

Forgetting to substitute fully when eliminating t.

Replace every t so the result is purely in x and y.

Key takeaways

x = f(t), y = g(t) give a curve via a parameter.
Eliminate t to get the Cartesian equation.
The range of t sets the domain and range of the curve.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A parametric curve expresses x and y in terms of:
- a parameter t
- each other
- a constant
- an angle only
Show answer and explanation
Answer: a parameter t
Third variable.
For $x = t+1$ , $y = t^2$ , the Cartesian equation is:
- $y = (x-1)^2$
- $y = x^2$
- $y = (x+1)^2$
- $y = x^2 - 1$
Show answer and explanation
Answer: $y = (x-1)^2$
t = x − 1.
To convert to Cartesian form you:
- eliminate t
- differentiate
- add the equations
- square both
Show answer and explanation
Answer: eliminate t
Remove the parameter.
The domain of the Cartesian curve is controlled by:
- the range of t
- the value of y
- the constant term
- nothing
Show answer and explanation
Answer: the range of t
Parameter restricts x.
For $x = 2t$ , $y = t$ , the Cartesian equation is:
- $y = \tfrac{x}{2}$
- $y = 2x$
- $y = x$
- $y = x^2$
Show answer and explanation
Answer: $y = \tfrac{x}{2}$
t = x/2 = y.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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