What a parameter does

In a parametric curve, both coordinates are functions of the same input.

For example,

does not say that y is directly a function of x at first. It says: choose a value of t, then calculate both coordinates. If t=-2, then

so the point is (-10,12). If t=0, the point is (0,0). If t=2, the point is (10,12). As t changes continuously, these points trace a curve.

[DIAGRAM: asset_name: Lesson p1.3.3: Parametric equations and Cartesian conversion - diagram 01; asset_slug: p1_3_3_understand_and_use_the_parametric_equations_of_curves_and_conversion_between_cartesian_and_parametric_forms__diagram_01; recommended_method: drawn_math; description: Draw a 16:9 two-panel diagram. The left panel shows a parameter line with points t=-2,-1,0,1,2 and the note that one t-value gives one coordinate pair. The right panel shows the curve from x=5t, y=3t^2 with the corresponding points labelled and arrows showing increasing t. Include the Cartesian result 25y=3x^2 as a direct label. Use NovaLearn style.]

The parameter is not a third coordinate. It is the input that generates the x-coordinate and the y-coordinate together. That is why a parametric equation can also show direction: increasing t tells you the order in which points are traced.

Worked example: find two points on the curve

When t=1,

so the point is (-2,3).

When t=-2,

so the point is (-5,9).

Eliminating a non-trigonometric parameter

To convert from parametric form to Cartesian form, remove the parameter. This is called eliminating the parameter.

The usual method is:

1. Rearrange one of the equations to make the parameter the subject.
2. Substitute that expression into the other equation.
3. Rearrange to give an equation involving only x and y.
4. Carry over any restrictions caused by the original parameter domain.

For the earlier example

the first equation gives

Substitute this into y=3t^2:

So

That is the Cartesian equation of the same parabola, provided the intended parameter values trace the whole curve.

Worked example: convert

to Cartesian form.

The x equation is linear in t, so it is the easiest one to rearrange:

Now substitute into the equation for y:

Simplify carefully:

so

To write this as a single fraction,

Expanding the numerator gives

The restriction t\ne 0 also matters. Since t=(x-2)/3, t\ne 0 means x\ne 2. The Cartesian form is therefore

The restriction is not decoration. Without it, the Cartesian equation might appear to allow a point that the original parameter can never produce.

Trigonometric parameters and circles

Some parametric equations use a trigonometric parameter, often written as theta or t. The key identity for circle conversion is

The basic circle

has centre (0,0) and radius r, because

so

For a circle with centre (a,b) and radius r, use

Worked example: convert

to Cartesian form.

First remove the translations and scale factor:

Now square both and add:

Multiply by 25:

This is a circle with centre (2,-4) and radius 5.

The common error is to use x^2+y^2=25 immediately. That would ignore the centre shift. Always rearrange to isolate cos t and sin t before using the identity.

Creating parametric forms

Conversion also works in the other direction. A Cartesian equation can often be written parametrically by choosing a useful parameter.

The simplest choice for a graph y=f(x) is usually

For example, the Cartesian curve

can be written as

This is not the only possible parametrisation. You could also choose x=3t, which would give

Both describe the same Cartesian curve if the parameter is allowed to range over all real values.

Worked example: give a parametric form for each curve.

For

choose

Then

For

choose

Then

so the equation is satisfied.

For the circle

use the centre-radius structure:

The parameter domain matters

The Cartesian equation tells you the shape. The parameter domain tells you which part of the shape is traced.

[DIAGRAM: asset_name: Lesson p1.3.3: Parametric equations and Cartesian conversion - diagram 02; asset_slug: p1_3_3_understand_and_use_the_parametric_equations_of_curves_and_conversion_between_cartesian_and_parametric_forms__diagram_02; recommended_method: drawn_math; description: Draw a 16:9 two-panel diagram for x=2+5cos t, y=-4+5sin t. The left panel shows 0<=t<=2pi tracing the whole circle with centre (2,-4) and radius 5. The right panel shows 0<=t<=pi tracing only the upper arc, with a faint dashed guide for the rest of the circle and labels t=0, t=pi/2, t=pi. Use NovaLearn style.]

For

the Cartesian equation is

If 0 <= t <= 2pi, the point traces the whole circle once. If 0 <= t <= pi, it traces only the upper semicircle:

• when t=0, the point is (7,-4);
• when t=pi/2, the point is (2,1);
• when t=pi, the point is (-3,-4).

So the same Cartesian equation can describe a whole circle, while the parametric equations plus their domain can describe a particular arc of that circle.

Worked example: describe the section of the curve

The Cartesian equation is still

But the parameter interval gives

So this is not the whole parabola. It is the section of

where

For this example, the y values also satisfy

but the most direct way to describe the section of the Cartesian curve is the x restriction, because x=5t is one-to-one on the interval.

The parameter domain matters Summary

That final distinction is the idea most worth checking in your own words.

Explain It Back

Use this as a self-explanation check after the section above. It is for diagnosing what you can already explain, not for learning new material from scratch.