AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
Unit 4A — Review
Notes — Concept Synthesis Across Topics 4.1 – 4.7
💡 Review Goals
By the end of this review you will be able to:
- Connect every skill in Unit 4A — parametric, implicit, and conic — into a unified framework
- Identify which topic a question is testing from its surface features
- Avoid the most common errors that AP graders flag in this unit
- Set up and execute a clean solution to a multi-step parametric or conic problem
1. The Unit at a Glance
Topic | Big Idea |
|---|---|
4.1 | Parametric form (x(t), y(t)); eliminate the parameter to get a rectangular equation. |
4.2 | Parametric motion: position, path, direction, self-intersections. |
4.3 | AROC of x and y; AROC of y w.r.t. x = slope of secant chord. |
4.4 | Parametric lines x = x₀ + at, y = y₀ + bt; circles x = h + r cos t, y = k + r sin t. |
4.5 | Implicit equations relate x and y; vertical line test for function status. |
4.6A | Parabolas: focus-directrix, (x−h)² = 4p(y−k) or (y−k)² = 4p(x−h). |
4.6B | Ellipses: sum of distances to foci is 2a; (x−h)²/a² + (y−k)²/b² = 1; c² = a² − b². |
4.6C | Hyperbolas: difference of distances; positive term gives transverse axis; c² = a² + b². |
4.7 | Parametrize implicit curves using matched trig identities. |
2. Key Formulas
Parametric:
- Line: (x₀ + at, y₀ + bt); slope b/a
- Circle radius r at (h, k): (h + r cos t, k + r sin t), t ∈ [0, 2π]
- Ellipse: (h + a cos t, k + b sin t)
- Hyperbola: (h + a sec t, k + b tan t)
Conic standard forms:
- Parabola: (x − h)² = 4p(y − k) or (y − k)² = 4p(x − h)
- Ellipse: (x − h)²/a² + (y − k)²/b² = 1, with a > b on the major axis
- Hyperbola: (x − h)²/a² − (y − k)²/b² = 1 (or with y first)
Distances:
- Ellipse: c² = a² − b²
- Hyperbola: c² = a² + b²
- Eccentricity: e = c/a (e < 1 ellipse, e = 1 parabola, e > 1 hyperbola)
3. Decision Tree for Conic Identification
Given an equation in general form Ax² + By² + Dx + Ey + F = 0:
- If only ONE squared term (just x² or just y²): PARABOLA
- If both squared terms with SAME sign and SAME coefficient: CIRCLE
- If both squared terms with SAME sign but DIFFERENT coefficients: ELLIPSE
- If squared terms with OPPOSITE signs: HYPERBOLA
4. Top AP-Style Pitfalls
⚠️ Common mistakes
- Forgetting that elimination of parameter loses information about direction and traced portion.
- Computing AROC of y w.r.t. x as Δy/Δt instead of Δy/Δx.
- Getting the sign of p wrong on a parabola: positive p opens up/right, negative opens down/left.
- Using c² = a² − b² for hyperbolas (it's c² = a² + b²).
- For ellipses, mistaking the larger denominator for b (it's actually a²).
- For hyperbolas, using the larger denominator to identify direction (it's actually the SIGN of each term).
- Forgetting that direction of a parametric circle depends on the sign convention of sin/cos.
5. Worked Mini-Example
📘 Example — Multi-step problem
A particle has position (x(t), y(t)) = (3 + 2 cos t, 4 + 5 sin t) for t ∈ [0, 2π].
- Eliminate the parameter: cos t = (x − 3)/2, sin t = (y − 4)/5; squared sum = 1, so (x − 3)²/4 + (y − 4)²/25 = 1
- Identify: ellipse, center (3, 4), a = 5 (vertical major axis), b = 2
- Vertices: (3, 9) and (3, −1); co-vertices (5, 4) and (1, 4)
- c² = 25 − 4 = 21, c = √21; foci (3, 4 ± √21)
- AROC of x on [0, π]: x(0) = 5, x(π) = 1, AROC = (1 − 5)/π = −4/π
- Particle moves left on average over [0, π] — consistent with the upper half of the ellipse.
6. Final Reminders
- State direction explicitly when describing parametric motion
- Show the elimination algebra completely on FRQ-style problems
- Always identify the conic type before applying formulas
- Complete the square systematically when converting from general form
- Check parametrizations by substitution before relying on them