AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.4 — Parametrically Defined Circles and Lines
Notes — Two Standard Parametric Curves
💡 Learning Objectives (4.4.A)
By the end of this lesson you will be able to:
- Write parametric equations for a line through two given points
- Write parametric equations for a circle (or arc) of given center and radius
- Identify the direction of travel and starting point from a parametric description
- Convert between rectangular and parametric forms for lines and circles
1. Parametric Lines
Given a starting point (x₀, y₀) and a direction vector (a, b) (the ‘velocity’ of motion), a line is parametrized by:
x(t) = x₀ + a t, y(t) = y₀ + b t
As t increases, the point (x(t), y(t)) moves along a straight line in the direction (a, b). At t = 0 the point is at (x₀, y₀).
📘 Example — Line through two points
Parametrize the line through (1, 2) and (4, 6).
- Direction vector: (4 − 1, 6 − 2) = (3, 4)
- Starting at (1, 2): x(t) = 1 + 3t, y(t) = 2 + 4t
- Check at t = 1: (4, 6) ✓
- Check at t = 0: (1, 2) ✓
2. Slope from a Parametric Line
For a line with parametric form x(t) = x₀ + at, y(t) = y₀ + bt:
slope = b/a (provided a ≠ 0)
Vertical lines have a = 0 (no change in x). Horizontal lines have b = 0 (no change in y) and slope 0.
3. Parametric Circles — Centered at Origin
A circle of radius r centered at the origin can be parametrized as:
x(t) = r cos t, y(t) = r sin t, t ∈ [0, 2π]
- At t = 0: (r, 0) — east
- At t = π/2: (0, r) — north
- At t = π: (−r, 0) — west
- At t = 3π/2: (0, −r) — south
- Direction: COUNTER-CLOCKWISE as t increases
4. Parametric Circles — General Center
To shift the circle to center (h, k):
x(t) = h + r cos t, y(t) = k + r sin t, t ∈ [0, 2π]
📘 Example — Circle centered at (2, −3) with radius 5
- x(t) = 2 + 5 cos t
- y(t) = −3 + 5 sin t
- At t = 0: (7, −3) — rightmost point
- At t = π: (−3, −3) — leftmost point
5. Reversing Direction or Choosing Starting Point
Modifying the parametrization changes how the circle is traced:
- To go CLOCKWISE, swap sin and cos roles or negate one: x = r cos t, y = −r sin t
- To start at the TOP, use: x = r cos(t + π/2) = −r sin t, y = r sin(t + π/2) = r cos t
- To trace HALF a circle, restrict t to [0, π] (upper half) or [π, 2π] (lower half)
- To trace the circle TWICE, let t go from 0 to 4π
- To speed up the trace, use cos(2t), sin(2t): then full circle traced for t ∈ [0, π]
6. Verifying a Parametric Description
Given a parametric description, you can confirm the curve type by eliminating the parameter:
📘 Example — Verify x = 1 + 4 cos t, y = 2 + 4 sin t is a circle
- From the equations: cos t = (x − 1)/4 and sin t = (y − 2)/4
- Square and add: ((x − 1)/4)² + ((y − 2)/4)² = cos²t + sin²t = 1
- So (x − 1)² + (y − 2)² = 16 — circle, center (1, 2), radius 4 ✓
7. Arcs of Circles
To parametrize an ARC (not a full circle), restrict t to a sub-interval. The endpoints of t correspond to the endpoints of the arc.
📘 Example — Quarter arc
Parametrize the arc of the unit circle from (1, 0) to (0, 1).
- Use x = cos t, y = sin t with t ∈ [0, π/2]
- At t = 0: (1, 0); at t = π/2: (0, 1) ✓
8. Summary
- Line through (x₀, y₀) with direction (a, b): x = x₀ + at, y = y₀ + bt
- Slope of parametric line: b/a (when a ≠ 0)
- Circle of radius r centered at (h, k): x = h + r cos t, y = k + r sin t, t ∈ [0, 2π]
- Counter-clockwise is the default; reverse by negating one trig function
- Restrict the t-interval to trace arcs or sub-paths