Parametric Equations

What is a parameter?

An independent variable (usually t) that both x and y depend on. It drives the curve but doesn't appear in the final equation after elimination.

What are parametric equations?

A pair of equations x = x(t) and y = y(t) that together describe a curve by expressing both coordinates in terms of a third variable t.

What is a parametric curve?

The set of all points (x(t), y(t)) traced out as t varies over a given interval. Also called a plane curve.

What is orientation?

The direction a point travels along the curve as t increases. Shown by arrows on the graph.

What does "eliminating the parameter" mean?

Removing t entirely to get a direct relationship between x and y only, such as y = 2x − 7 or x²/9 + y²/9 = 1.

How do you eliminate the parameter from algebraic equations?

Solve one equation for t, then substitute that expression into the other equation.

Which equation should you solve for t first?

Whichever looks simpler to isolate t from. There is no strict rule — just pick the easier one.

How do you eliminate the parameter from trig equations?

Do NOT solve for t. Instead isolate cos t and sin t separately, then substitute into the identity cos²t + sin²t = 1.

Why don't you solve for t in trig problems?

Because solving for t would require inverse trig (like arccos) which gets messy. The Pythagorean identity lets you skip t entirely.

What is the Pythagorean trig identity used in parametric equations?

cos²t + sin²t = 1. After isolating cos t and sin t from x and y, substitute both into this identity to eliminate t.

What are the parametric equations for a circle of radius r centered at the origin?

x(t) = r cos t, y(t) = r sin t, where 0 ≤ t ≤ 2π.

What is the rectangular form of a circle with radius r?

x² + y² = r². You get this by eliminating the parameter from x = r cos t, y = r sin t.

What are the parametric equations for an ellipse?

x(t) = a cos t, y(t) = b sin t, where a and b are the two semi-axes.

What is the rectangular form of an ellipse?

x²/a² + y²/b² = 1. The semi-major axis is the larger of a and b, and the semi-minor axis is the smaller.

Where do the semi-axis values a and b come from in the ellipse equation?

Square root the denominators. From x²/25 + y²/4 = 1, take √25 = 5 and √4 = 2.

What is the semi-major axis?

The longer radius of an ellipse — whichever of a or b is larger. The ellipse stretches furthest in that direction.

What is parameterizing a curve?

Starting from a single equation like y = f(x) and writing it as two parametric equations x(t) and y(t).

v]What is the simplest way to parameterize y = f(x)?

Let x(t) = t, then replace every x in the equation with t to get y(t) = f(t).

Is there only one correct parameterization of a curve?

No — there are infinitely many valid parameterizations. You have total freedom in choosing x(t) as long as its range covers the original domain.

What is a cycloid?

The path traced by a point on the rim of a rolling wheel. Its parametric equations are x(t) = a(t − sin t), y(t) = a(1 − cos t), where a is the wheel's radius and t is the angle rotated.

What is the difference between a cycloid and a hypocycloid?

A cycloid is traced by a point on a wheel rolling along a flat surface. A hypocycloid is traced by a point on a small circle rolling inside a larger circle.

What shape do you get when both x(t) and y(t) are linear in t?

A straight line.

Common mistake — what goes wrong when you write x/25 + y/4 = 1 instead of x²/25 + y²/4 = 1?

You forgot to square after substituting. (x/5)² = x²/25, not x/25. Always square the entire expression.

Common mistake — when parameterizing with x(t) = 2t, what is (2t)²?

4t², not 2t². You must square both the coefficient and the variable — squaring only t is a common error.