Topic A: Features of Quadratic Functions
Lessons 1–5 · Vertex, axis of symmetry, intercepts.
01–03 — Lessons 1–3
Key Features of Parabolas
- Vertex (h, k): highest or lowest point.
- Axis of symmetry: x = h, a vertical line through the vertex.
- Parabola opens up (a > 0) → minimum. Opens down (a < 0) → maximum.
- Y-intercept: f(0) = c in standard form ax² + bx + c.
- X-intercepts (zeros): where f(x) = 0.
f(x) = 2(x−3)² + 1: vertex (3,1), axis x=3, opens up (minimum at 1).
Practice — Lessons 1–3
Q1 · Multiple Choice
For f(x) = −(x+2)² + 5, what is the vertex?
Q2 · Multiple Choice
A parabola opens downward. The vertex is a:
04–05 — Lessons 4–5
Vertex Form & Standard Form
- Vertex form: f(x) = a(x − h)² + k → vertex (h, k).
- Standard form: f(x) = ax² + bx + c → vertex at x = −b/(2a).
- Convert vertex → standard by expanding.
- Convert standard → vertex by completing the square or using x = −b/(2a).
- Use vertex form for easy graphing; standard form for finding y-intercept.
f(x) = x² − 6x + 8: vertex x = −(−6)/(2·1) = 3 → f(3) = 9−18+8 = −1 → vertex (3,−1).
Practice — Lessons 4–5
Q3 · Multiple Choice
Find the axis of symmetry of f(x) = 2x² − 8x + 3.
Q4 · Short Answer
For f(x) = x² + 4x + 7, what is f(0)?
Topic B: Analyzing Quadratics in Context
Lessons 6–9 · Modeling projectile motion and other quadratic situations.
06–09 — Lessons 6–9
Quadratics in Context
Quadratics model real-world situations like projectile motion, area, and profit.
- Height of a projectile: h(t) = −16t² + v₀t + h₀ (in feet).
- Vertex = maximum height and time at which it occurs.
- X-intercepts = when the object hits the ground (h = 0).
- Y-intercept = initial height (t = 0).
- Negative coefficient means gravity pulls down.
h(t) = −16t² + 64t → vertex at t=2: h(2) = −64+128 = 64 ft max height. Lands at t=4.
Practice — Lessons 6–9
Q5 · Multiple Choice
For h(t) = −16t² + 64t, when does the ball reach max height?
Q6 · Short Answer
Using h(t) = −16t² + 64t, what is the maximum height?