Choosing the parameter is often the hardest part of setting up a parametric equation. Fortunately, there are some simple rules that make the process much easier.

The main idea is:

Choose a parameter that makes x(t) and y(t) as simple as possible and describes the required part of the curve without ambiguity.

Here are some practical strategies.

1. First Try t = x

If the curve is already given as:

$y=f(x)$

the easiest choice is usually:

$t=x$

Then:

$x=t$

and immediately:

$y=f(t)$

Example

For:

$y=x^2$

choose:

$t=x$

Then:

$x=t$

and:

$y=t^2$

This is probably the simplest possible parameterization.

Tip

If a problem gives y as a function of x, always consider t = x first.

2. Try t = y When x Is Easier to Express in Terms of y

Sometimes the equation is naturally solved for x.

Example

Consider:

$y=x^2$

Suppose the right half of the parabola is required.

Choose:

$t=y$

Then:

$y=t$

Since:

$t=x^2$

we obtain:

$x=\sqrt{t}$

Therefore:

$x(t)=\sqrt{t}$ $y(t)=t$

This may be particularly useful when the calculation involves dy rather than dx.

3. If the Curve Is Given as x = f(y), Try t = y

Suppose:

$x=y^3$

The natural choice is:

$t=y$

Therefore:

$y=t$

and:

$x=t^3$

So:

$x(t)=t^3$ $y(t)=t$

There is no need to do any complicated algebra.

4. For Circles, Think of Angles

A circle is a classic example where choosing t = x or t = y is not ideal for describing the entire curve.

Consider:

$x^2+y^2=a^2$

The standard parameterization is:

$x=a\cos t$ $y=a\sin t$

Here, t is the angle made by the radius with the positive x-axis.

As t goes from 0 to 2π, the point travels around the entire circle.

This is much better than trying to use x or y as the parameter because a vertical or horizontal line can intersect a circle at two points.

Tip

For circles and many problems involving trigonometry:

Think of t as an angle.

5. For Ellipses, Use the Same Idea

Consider:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The natural parameterization is:

$x=a\cos t$ $y=b\sin t$

Again, t represents an angle-like parameter.

This works because:

$\cos^2t+\sin^2t=1$

Substituting gives:

$\frac{(a\cos t)^2}{a^2}+\frac{(b\sin t)^2}{b^2}=1$

which simplifies to:

$\cos^2t+\sin^2t=1$

So the equation of the ellipse is automatically satisfied.

6. For Lines, Let t Control Movement Along the Line

Suppose a line passes through:

(x₁, y₁)

with direction vector:

(a, b)

A natural parameterization is:

$x=x_1+at$ $y=y_1+bt$

For example, a line through (2, 3) with direction (4, 1) can be written:

$x=2+4t$ $y=3+t$

As t changes, the point moves along the line.

Tip

For straight lines:

Use t to represent how far the point has travelled along the line.

7. Look for Expressions That Suggest Trigonometric Identities

Sometimes the equation contains expressions such as:

$x^2+y^2$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}$

These often suggest sine and cosine.

For example:

$x^2+y^2=25$

suggests:

$x=5\cos t$ $y=5\sin t$

because:

$\cos^2t+\sin^2t=1$

This is usually much easier than solving the equation for y.

8. Use t to Simplify Radicals

Suppose:

$y=\sqrt{x}$

The expression √x suggests choosing:

$x=t^2$

Then:

$y=\sqrt{t^2}$

For t ≥ 0:

$y=t$

So instead of:

$x=t$ $y=\sqrt{t}$

we get the much cleaner parameterization:

$x=t^2$ $y=t$

Tip

When you see a square root, consider choosing a parameter that eliminates the square root.

9. Use t to Simplify Powers

Suppose:

$y=x^{1/3}$

You could choose:

$x=t$

giving:

$y=t^{1/3}$

But another convenient choice is:

$y=t$

Then:

$x=t^3$

So:

$x=t^3$ $y=t$

The second version can be much easier in some calculations.

10. Look at What the Problem Asks You to Calculate

This is one of the most useful practical tips.

The “best” parameterization isn’t necessarily the one with the shortest equations.

It may be the one that makes the calculation easiest.

For example, area under a parametric curve can be written as:

$A=\int y(t)\frac{dx}{dt},dt$

So a parameterization that makes y(t) and dx/dt simple is desirable.

For surface area, arc length, or other calculations, a different parameterization might be more convenient.

Example

Suppose:

$y=x^2$

and the required area involves integrating with respect to x.

Choosing:

$x=t$

gives:

$y=t^2$

and:

$\frac{dx}{dt}=1$

which is extremely simple.

11. Check Whether the Parameter Covers the Required Curve

A parameterization might be mathematically correct but still cover only part of the curve.

For example:

$x=\sqrt{t}$ $y=t$

describes the right-hand portion of:

$y=x^2$

because x is always non-negative.

It does not describe the left-hand portion.

To describe the left side, use:

$x=-\sqrt{t}$ $y=t$

Therefore, always ask:

Which part of the curve does this parameterization actually cover?

12. Check Whether One Value of t Gives Only One Point

A good parameterization should normally give a definite point for each value of t.

For example, on the circle:

$x^2+y^2=1$

using t = x creates a problem if we try to write y as a single function of t:

$y=\pm\sqrt{1-t^2}$

There are two possible y-values for most x-values.

Using:

$x=\cos t$ $y=\sin t$

solves this problem elegantly.

A Useful Decision-Making Checklist

When asked to parameterize a curve, try these steps.

Step 1: Look at the Equation

Is it already:

$y=f(x)$

Try:

$t=x$

Step 2: Can It Easily Be Written as x = f(y)?

If so, try:

$t=y$

Step 3: Is There a Square Root?

Try setting the quantity under the square root equal to something like t².

For example, if:

$y=\sqrt{x}$

try:

$x=t^2$

Step 4: Is It a Circle or Ellipse?

Think:

$x=a\cos t$ $y=b\sin t$

Step 5: Is It a Line?

Let t represent movement along the line.

Step 6: Are There Powers or Radicals That Can Be Simplified?

Choose t to eliminate them.

Step 7: Consider the Calculation

Ask whether the parameterization makes:

derivatives simpler,
integrals simpler,
limits simpler,
or the required geometry easier to describe.

Step 8: Check the Coverage

Make sure the chosen parameter describes the entire portion of the curve required by the problem.

Also check whether the parameterization introduces multiple points for the same value of t or misses part of the curve.

The Big Picture

There is no universal formula for choosing x(t).

Instead, parameterization is a bit like choosing the best coordinate system for a particular job.

For a parabola:

t = x may be easiest.

For a right-hand parabola:

t = y may be convenient.

For a circle:

t = angle is usually best.

For a line:

t = distance/direction of travel is natural.

For radicals:

t can be chosen to eliminate the radical.

The ultimate goal is simple:

Choose t so that the curve becomes easy to describe and the subsequent mathematics becomes easier to perform.

Tips for Choosing a Parameter in Parametric Equations

1. First Try t = x

Example

Tip

2. Try t = y When x Is Easier to Express in Terms of y

Example

3. If the Curve Is Given as x = f(y), Try t = y

4. For Circles, Think of Angles

Tip

5. For Ellipses, Use the Same Idea

6. For Lines, Let t Control Movement Along the Line

Tip

7. Look for Expressions That Suggest Trigonometric Identities

8. Use t to Simplify Radicals

Tip

9. Use t to Simplify Powers

10. Look at What the Problem Asks You to Calculate

Example

11. Check Whether the Parameter Covers the Required Curve

12. Check Whether One Value of t Gives Only One Point

A Useful Decision-Making Checklist

Step 1: Look at the Equation

Step 2: Can It Easily Be Written as x = f(y)?

Step 3: Is There a Square Root?

Step 4: Is It a Circle or Ellipse?

Step 5: Is It a Line?

Step 6: Are There Powers or Radicals That Can Be Simplified?

Step 7: Consider the Calculation

Step 8: Check the Coverage

The Big Picture

Like this:

DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series

1. First Try t = x

Example

Tip

2. Try t = y When x Is Easier to Express in Terms of y

Example

3. If the Curve Is Given as x = f(y), Try t = y

4. For Circles, Think of Angles

Tip

5. For Ellipses, Use the Same Idea

6. For Lines, Let t Control Movement Along the Line

Tip

7. Look for Expressions That Suggest Trigonometric Identities

8. Use t to Simplify Radicals

Tip

9. Use t to Simplify Powers

10. Look at What the Problem Asks You to Calculate

Example

11. Check Whether the Parameter Covers the Required Curve

12. Check Whether One Value of t Gives Only One Point

A Useful Decision-Making Checklist

Step 1: Look at the Equation

Step 2: Can It Easily Be Written as x = f(y)?

Step 3: Is There a Square Root?

Step 4: Is It a Circle or Ellipse?

Step 5: Is It a Line?

Step 6: Are There Powers or Radicals That Can Be Simplified?

Step 7: Consider the Calculation

Step 8: Check the Coverage

The Big Picture

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Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series