Precalculus

Parametric Equations

So far, you’ve written equations as y = f(x) — y depends directly on x. But what if both x and y depend on a third variable? That variable is called a parameter, usually written as t. Welcome to parametric equations.

Part 1: What Are Parametric Equations?

Instead of one equation, you get two:

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

As t changes, the point (x, y) traces out a curve. Think of t as time: at each moment, you have a position (x, y).

A Simple Example: A Straight Line

x = t, \quad y = 2t + 1

This is just y = 2x + 1 in disguise. But parametric form lets us do much more.

Part 2: Circles — Where Parametric Shines

Try writing a circle as y = f(x). You can’t do it with a single function (you’d need the top half and bottom half separately). But parametrically, a circle is elegant:

x = r\cos(t), \quad y = r\sin(t)

As t goes from 0 to 2pi, the point traces a full circle of radius r.

Radius (r)3

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The green curve shows x(t) = rcos(t), and the red shows y(t) = rsin(t). Together they create circular motion.

Try This

Why cos and sin? Remember the unit circle: at angle t, the point on the circle is (cos t, sin t). Multiplying by r just scales the circle up. Drag the radius slider and watch all curves scale together.

Part 3: Ellipses — Stretched Circles

An ellipse is a circle that’s been stretched differently in x and y:

x = a\cos(t), \quad y = b\sin(t)

When a = b, you get a circle. When a and b differ, the shape stretches.

a (horizontal radius)4

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b (vertical radius)2

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Connection

Planets orbit in ellipses! Kepler discovered that planetary orbits are elliptical, not circular. The Sun sits at one focus of the ellipse. Parametric equations are exactly how astronomers describe these orbits.

Part 4: Speed Along the Curve

The parameter t often represents time. How fast t changes affects how fast you move along the curve. Compare these two parameterizations of the same line:

\text{Slow: } x = t, \; y = t \quad \text{vs.} \quad \text{Fast: } x = 3t, \; y = 3t

Both trace the same path (y = x), but the second one covers it three times as fast. The direction can change too:

Speed multiplier1

-33

Try This

Set speed to -1. Now the point traces the same line but in the opposite direction. Parametric equations encode not just the shape of a curve, but the direction and speed of travel along it.

Part 5: Lissajous Figures — Fancy Parametric Art

When both x and y are sinusoidal with different frequencies, you get beautiful patterns called Lissajous figures:

x = \sin(at), \quad y = \sin(bt)

Frequency a (for x)1

Frequency b (for y)2

When a = 1 and b = 2, the y-component oscillates twice as fast as x. The combined (x, y) path creates a figure-eight. Different frequency ratios create different patterns — try a = 3, b = 2 for a pretzel shape!

Challenge

Challenge:

Set a = 1 and b = 1. What shape do you expect? (Hint: same frequency means the point oscillates back and forth on a line.)
Set a = 2 and b = 3. How many “loops” does the resulting Lissajous figure have?
Can you find settings that create a pattern with exactly 4 crossings?

Wrapping Up

Concept	Key Idea
Parametric equations	x and y both depend on a parameter t
Circle	x = r cos(t), y = r sin(t)
Ellipse	x = a cos(t), y = b sin(t)
Direction & speed	Encoded in how t maps to (x, y)
Lissajous figures	Different frequencies create intricate patterns

Parametric equations let you describe curves that regular y = f(x) can’t handle. They’re essential in physics (projectile motion), engineering (CNC machining paths), and computer graphics (Bezier curves). Once you think parametrically, a whole new world of curves opens up.

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