AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.1 — Parametric Functions
Notes — Describing Curves with a Parameter
💡 Learning Objectives (4.1.A)
By the end of this lesson you will be able to:
- Define a parametric function as a pair (x(t), y(t)) describing a curve in the plane
- Construct a table of values and sketch a parametric curve
- Recognize that a parametric description carries DIRECTION as well as shape
- Convert simple parametric equations into a single rectangular equation by eliminating the parameter
1. What Is a Parametric Function?
Up to now, every function you've graphed has been of the form y = f(x) — for each input x there is one output y. But many curves in the plane fail the vertical line test: circles, ellipses, figure-eights, loops. To handle these, we introduce a third variable, called the PARAMETER, that drives both x and y.
A parametric function is a pair of equations x = x(t) and y = y(t), where the parameter t ranges over an interval.
As t varies through its interval, the point (x(t), y(t)) traces out a curve. The variable t is often time, but it doesn't have to be — it can be any quantity we sweep through.
2. Building a Curve from a Table
📘 Example — Sketch x(t) = t², y(t) = t for t ∈ [−2, 2]
Build a table:
- t = −2: (4, −2)
- t = −1: (1, −1)
- t = 0: (0, 0)
- t = 1: (1, 1)
- t = 2: (4, 2)
Plot the points and connect with a smooth curve. The result is a sideways parabola opening to the right.
3. Direction Matters
Unlike y = f(x), a parametric curve has a DIRECTION OF MOTION baked in. As t increases, the point (x(t), y(t)) traces the curve in a specific order. This direction (called orientation) is part of the parametric description — and is often indicated by an arrow on the sketch.
⚠️ Direction-related caution
Two different parametric descriptions can trace the SAME curve in the plane but go in opposite directions, or trace different portions of it. The pair (x(t), y(t)) is more information than just a sketch.
4. Eliminating the Parameter
Sometimes it's useful to find a relationship between x and y that does NOT involve t. The standard method:
- Solve one of the equations for t in terms of x or y
- Substitute that expression into the other equation
- Simplify
📘 Example — Eliminate the parameter from x = 2t + 1, y = t²
- Solve the first for t: t = (x − 1)/2
- Substitute into the second: y = ((x − 1)/2)² = (x − 1)²/4
- So in rectangular form: y = (x − 1)²/4 — a parabola with vertex (1, 0)
⚠️ Loss of information
Eliminating the parameter often loses information about direction AND about the portion of the curve actually traced. If t ∈ [0, 5], only part of the rectangular curve is covered. Always note any restrictions.
5. When the Parameter Is Trigonometric
Many of the most useful parametric curves use sin and cos as the parameter functions. The Pythagorean identity is the key tool for elimination:
📘 Example — Eliminate from x = 3 cos t, y = 3 sin t
- From x = 3 cos t: cos t = x/3
- From y = 3 sin t: sin t = y/3
- Use sin²t + cos²t = 1: (x/3)² + (y/3)² = 1, i.e. x² + y² = 9
- So this is a circle of radius 3 centered at the origin.
6. Reading Information from the Parametric Form
The parametric form often makes certain features easy to read, even before eliminating t:
- x-intercepts of the curve: solve y(t) = 0
- y-intercepts: solve x(t) = 0
- Starting and ending points: evaluate (x, y) at the endpoints of the t-interval
- Self-intersections: find different t-values that give the same (x, y)
7. A Numerical View
t | 0 | π/4 | π/2 | 3π/4 | π | 5π/4 | 3π/2 | 7π/4 | 2π |
|---|---|---|---|---|---|---|---|---|---|
x = cos t | 1 | √2/2 | 0 | −√2/2 | −1 | −√2/2 | 0 | √2/2 | 1 |
y = sin t | 0 | √2/2 | 1 | √2/2 | 0 | −√2/2 | −1 | −√2/2 | 0 |
As t increases from 0 to 2π, the point (cos t, sin t) traces the unit circle counter-clockwise once.
8. Summary
- Parametric form (x(t), y(t)) describes a curve in the plane via a parameter t
- Each value of t produces ONE point on the curve, plotted in order
- Direction of travel is part of the description; arrows on a sketch show this
- Eliminate the parameter by solving one equation for t and substituting into the other
- For trigonometric parameters, sin²t + cos²t = 1 is the elimination tool