AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.7 — Parametrization of Implicitly Defined Functions
Notes — Turning Implicit Curves into Parametric Form
💡 Learning Objectives (4.7.A)
By the end of this lesson you will be able to:
- Find a parametric description of a curve given its implicit equation
- Use trigonometric substitutions to parametrize circles, ellipses, and hyperbolas
- Verify a parametric description matches the implicit equation
- Recognize that a single implicit curve has many parametrizations
1. Why Parametrize?
An implicit equation describes a curve in the plane, but it doesn't directly tell you how to MOVE along the curve or how to describe it as a function of a single variable. A parametric description does both — making it easy to (a) trace the curve in code or by hand, (b) handle curves that fail the vertical line test, and (c) study motion along the curve.
2. Parametrizing Circles
A circle (x − h)² + (y − k)² = r² is naturally parametrized using sine and cosine, because cos²t + sin²t = 1 matches the form of the equation:
x = h + r cos t, y = k + r sin t
To verify: substitute into the implicit equation. ((h + r cos t) − h)² + ((k + r sin t) − k)² = r² cos²t + r² sin²t = r²(cos²t + sin²t) = r² ✓
3. Parametrizing Ellipses
An ellipse (x − h)²/a² + (y − k)²/b² = 1 is parametrized using the same trick, but with the radii baked in:
x = h + a cos t, y = k + b sin t
📘 Example — Parametrize x²/25 + y²/9 = 1
- Center (0, 0), a = 5, b = 3
- Parametric: x = 5 cos t, y = 3 sin t for t ∈ [0, 2π]
- Verify: (5 cos t)²/25 + (3 sin t)²/9 = cos²t + sin²t = 1 ✓
4. Parametrizing Hyperbolas
Hyperbolas are parametrized using the HYPERBOLIC functions or, more simply, with secant and tangent. Because sec²t − tan²t = 1, we can match a hyperbola's form (x − h)²/a² − (y − k)²/b² = 1:
x = h + a sec t, y = k + b tan t (right branch when sec t > 0)
📘 Example — Parametrize x²/4 − y²/9 = 1
- a = 2, b = 3, center origin
- x = 2 sec t, y = 3 tan t
- Verify: (2 sec t)²/4 − (3 tan t)²/9 = sec²t − tan²t = 1 ✓
5. Parametrizing Parabolas
Parabolas, being functions of a single variable, parametrize trivially. For y = f(x), let t = x:
x = t, y = f(t)
📘 Example — Parametrize y = x²
- x = t, y = t² for t ∈ ℝ
- This traces the parabola from far-left (t large negative) through the origin (t = 0) to far-right (t large positive).
6. Different Parametrizations of the Same Curve
There is no UNIQUE parametrization. The same circle can be described many ways:
- (cos t, sin t) for t ∈ [0, 2π]: counter-clockwise once
- (cos(2t), sin(2t)) for t ∈ [0, π]: faster, but still once
- (cos t, −sin t) for t ∈ [0, 2π]: clockwise
- (sin t, cos t) for t ∈ [0, 2π]: counter-clockwise but starting at the top
The parametrization choice can affect direction, speed, starting point, and which portion of the curve is traced.
7. Verifying a Parametrization
Given a candidate parametrization for an implicit equation:
- Substitute x(t) and y(t) into the implicit equation
- Simplify and check that the result is the constant on the right side
- Confirm that the t-interval covers the desired portion of the curve
📘 Example — Verify
Verify that x = 1 + 3 cos t, y = 2 − 3 sin t parametrizes (x − 1)² + (y − 2)² = 9.
- (1 + 3 cos t − 1)² + (2 − 3 sin t − 2)² = 9 cos²t + 9 sin²t = 9 ✓
- Note: the negative sign on sin reverses direction (clockwise instead of counter-clockwise).
8. Other Implicit Curves
Not every implicit curve has a clean parametrization. For curves like x³ + y³ = 6xy (folium of Descartes), parametrization requires creative choices (e.g., y = tx). The Pythagorean-style trick we use for circles and conics depends on having a recognizable trig identity to match the equation.
9. Summary
- Parametrize using identities matched to the implicit equation
- Circle / ellipse: x = h + a cos t, y = k + b sin t
- Hyperbola: x = h + a sec t, y = k + b tan t
- Parabola y = f(x): x = t, y = f(t)
- A given curve has many parametrizations; check direction, speed, and portion covered
- Verify any parametrization by substituting into the implicit equation