C3PureStretch

Parametric Equations

Q: What is a parameter?

A third variable (usually t) that both x and y depend on. As it varies, it generates the points of the curve one at a time — often interpreted as time driving a moving point.

Q: How do I convert parametric equations to Cartesian form?

Eliminate the parameter. Rearrange one equation to make t the subject and substitute into the other, or, for trigonometric parameters, use ^2 t + ^2 t = 1 to remove t.

Q: Does the parameter range matter?

Yes. If t is restricted, only part of the Cartesian curve is traced, so the domain or range of the final curve may be limited. Always carry the restriction through.

Instead of linking $y$ to $x$ directly, parametric equations give both coordinates in terms of a third variable — a parameter, often $t$ . It is the natural language for motion and for curves that no single $y = f (x)$ can describe.

28 min Video by TLMaths Coordinate Geometry

The big picture

Some curves simply cannot be written as $y = f (x)$ : a full circle fails the vertical line test, and the path of a projectile is more naturally described by “where is it at time $t$ ?”. Parametric equations solve this by letting a parameter drive both coordinates independently, tracing the curve out point by point as the parameter changes. This is exactly how motion works — position as a function of time — which is why parametrics underpin projectile mechanics and are differentiated with the chain rule in Year 2. The core skills are converting to and from ordinary (Cartesian) form and respecting the restrictions the parameter imposes.

A-Level Maths C3-01: Introducing Parametric EquationsWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Interpret a pair of parametric equations as a curve traced by a parameter
Find coordinates of points from parameter values
Convert parametric equations to Cartesian form by eliminating the parameter
Understand how the parameter’s range restricts the curve

What parametric form means

A curve is given $parametrically$ when $x$ and $y$ are each written as functions of a parameter, for example $x = t + 1$ , $y = t^{2}$ . Feed in a value of $t$ and you get one point $(x, y)$ ; sweep $t$ through its range and the points trace out the curve.

Think of $t$ as time and the curve as a path: at each instant the object is at a definite spot, and the parameter records the order in which the curve is drawn — information a plain $y = f (x)$ throws away.

1Substitute

t = 3

into each equation.

x = 3 + 1 = 4

and

y = 3^{2} = 9

AnswerThe point

(4, 9)

Why this works

The same $(x, y)$ curve can be traced by many different parametrisations, at different speeds or directions. The parameter carries extra information — how the curve is traversed — beyond just its shape.

Converting to Cartesian form

To find the ordinary equation of the curve, $eliminate the parameter$ : make $t$ the subject of one equation and substitute into the other, leaving a relation purely between $x$ and $y$ .

When trigonometric parameters appear (e.g. $x = cos t$ , $y = sin t$ ), you usually cannot solve for $t$ neatly — instead use an identity such as $sin^{2} t + cos^{2} t = 1$ to eliminate it in one move.

x = f (t), y = g (t) eliminate t relation in x, y

Rearrange one equation for

t

, substitute into the other.

1From the first equation,

t = x - 1

2Substitute into

y = t^{2}

y = (x - 1)^{2}

Answer

y = (x - 1)^{2}

— a parabola.

Tip — For trigonometric parameters, reach for the identity $sin^{2} t + cos^{2} t = 1$ rather than trying to solve for $t$ directly.

The parameter’s range restricts the curve

The values the parameter is allowed to take control which part of the curve you actually get. If $t$ only runs over a limited interval, the Cartesian curve is a $restricted$ piece — so the domain and range of the final curve may be smaller than the full Cartesian equation suggests.

Always translate any restriction on $t$ into the corresponding restriction on $x$ (and $y$ ). This is a frequent source of dropped marks: the tidy Cartesian equation can quietly overstate the curve.

1Eliminate

t

t = y

, so

x = y^{2}

2But

t \geq 0

means

y = t \geq 0

3So only the upper half of the sideways parabola

x = y^{2}

is traced.

Answer

x = y^{2}

with

y \geq 0

— the top half only.

Think like an examiner

State any restriction on $x$ or $y$ that follows from the parameter’s range — the Cartesian equation alone is often not the full answer.
For trigonometric parametrisations, use $sin^{2} t + cos^{2} t = 1$ to eliminate $t$ cleanly.
When finding points, substitute the parameter value into $both$ equations — a point needs both coordinates.
Show the elimination step clearly; “convert to Cartesian form” is marked on the method, not just the final equation.

Common misconceptions

Giving the Cartesian equation but ignoring the parameter’s range.

Carry restrictions through: a limited

t

gives a limited portion of the curve.

Trying to solve

x = cos t

for

t

and substituting.

Use the identity

sin^{2} t + cos^{2} t = 1

to eliminate

t

in one step.

Finding only

x

(or only

y

) for a given

t

A point has both coordinates — substitute

t

into each equation.

Assuming each Cartesian curve has a unique parametric form.

Many parametrisations give the same curve; the parameter adds information about how it is traced.

Parametric essentials

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

Both coordinates via a parameter.

eliminate t \to Cartesian

Substitute one into the other.

sin^{2} t + cos^{2} t = 1

Eliminating a trig parameter.

Stretch yourself

A curve is given by $x = 2 t$ , $y = t^{2} - 1$ . Find its Cartesian equation, and the coordinates where it crosses the $x$ -axis.

Hint — Eliminate $t$ using $t = \frac{x}{2}$ . The curve crosses the $x$ -axis where $y = 0$ .

Questions students ask

What is a parameter?

A third variable (usually

t

) that both

x

and

y

depend on. As it varies, it generates the points of the curve one at a time — often interpreted as time driving a moving point.

How do I convert parametric equations to Cartesian form?

Eliminate the parameter. Rearrange one equation to make

t

the subject and substitute into the other, or, for trigonometric parameters, use

sin^{2} t + cos^{2} t = 1

to remove

t

Why use parametric equations at all?

They describe curves that are not functions (like a full circle) and capture motion naturally as position over time. They are essential for projectiles in mechanics and for curves you later differentiate with the chain rule.

Does the parameter range matter?

Yes. If

t

is restricted, only part of the Cartesian curve is traced, so the domain or range of the final curve may be limited. Always carry the restriction through.

Key takeaways

Parametric form gives $x$ and $y$ each as functions of a parameter $t$ .
Substitute a $t$ -value into both equations to get a point.
Convert to Cartesian by eliminating $t$ (or using a trig identity).
The parameter’s range restricts which part of the curve appears.
Parametrics describe motion and non-function curves — key for mechanics and later calculus.

How this fits the course

Build on

Straight-Line Graphs
Trig identities (later)soon

Circles

Leads to

Arithmetic Sequences and Series
Projectiles — Mechanics (later)soon

Test yourself

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