Overview
- Topic: Graphing quadratic functions in vertex and standard form.
- Goals: Find vertex, axis of symmetry, max/min, domain, range, intercepts, and write equations from graphs.
- Includes techniques: plotting using symmetry, tables centered at vertex, completing the square, quadratic formula, and word problem application.
Key Concepts
- Quadratic general forms:
- Vertex form: y = a(x − h)^2 + k, vertex = (h, k).
- Standard form: y = ax^2 + bx + c.
- Parabola direction:
- a > 0: opens upward, has a minimum at vertex.
- a < 0: opens downward, has a maximum at vertex.
- Axis of symmetry: vertical line x = h (x-coordinate of vertex).
- Domain of any quadratic: (−∞, ∞).
- Range depends on vertex and opening:
- Upward: [k, ∞).
- Downward: (−∞, k].
Graphing From Vertex Form
- Identify h and k from y = a(x − h)^2 + k to get vertex (h,k).
- Use symmetry: choose points one and two units left and right from vertex.
- For a = 1: move 1 right → y increases by 1, move 2 right → y increases by 4.
- For a ≠ 1: vertical changes scale by |a| (e.g., a = −2 doubles vertical changes and reflects).
- Table method: center x-values around the vertex; pick two points each side.
- Intercepts:
- Y-intercept: set x = 0, evaluate y.
- X-intercepts: set y = 0, solve for x (factor or use algebraic steps).
- Examples summarized:
- y = (x − 1)^2: vertex (1,0), AOS x = 1, min 0, domain (−∞,∞), range [0,∞).
- y = x^2 + 4: vertex (0,4), opens up, min 4, x-intercepts ±2, range [4,∞).
- y = (x + 2)^2 − 1: vertex (−2,−1), AOS x = −2, min −1, x-intercepts −3 and 1, y-intercept 3.
- y = −2(x − 1)^2 + 3: vertex (1,3), opens down, max 3, vertical scale doubles changes, x-intercepts found via algebra.](streamdown:incomplete-link)
Graphing From Standard Form
- Vertex x-coordinate: x = −b / (2a).
- Find y by substituting x into the function.
- X-intercepts: factor ax^2 + bx + c if possible, or use quadratic formula.
- Y-intercept: y = c (plug x = 0).
- Use symmetry: vertex x is midpoint of x-intercepts.
- Example workflow:
- y = x^2 + 2x − 8:
- a = 1, b = 2, c = −8.
- Vertex x = −2/2 = −1; y = 9 → vertex (−1,9).
- Factor to find x-intercepts: (x + 4)(x − 2) → x = −4, 2.
- Y-intercept: (0,−8).
- Domain (−∞,∞), range [9,∞).
- y = x^2 + 2x − 3:
- X-intercepts: −3 and 1.
- Vertex (−1,−4) found by averaging intercepts or −b/(2a).
- Complete the square to rewrite in vertex form: (x + 1)^2 − 4.](streamdown:incomplete-link)
Completing The Square (Convert Standard → Vertex Form)
- Steps:
- Group first two terms: ax^2 + bx + c → if a ≠ 1, factor a from first two terms.
- Add and subtract (b/2a)^2 inside to form perfect square.
- Balance equation by adjusting constant term.
- Factor perfect square → vertex form a(x − h)^2 + k.
- Useful to read vertex directly and to graph.
Quadratic Formula (Solve For X)
- Use when factoring is hard: x = [−b ± sqrt(b^2 − 4ac)] / (2a).
- Apply to find x-intercepts (real solutions correspond to x-intercepts).
- Discard negative-time solutions in real-time word problems.
Structured Summary Table (Examples Compared)
| Function | Vertex (h,k) | AOS | Opens | Intercepts (x,y) | Domain | Range |
|---|
| y = (x − 1)^2 | (1, 0) | x = 1 | Up | y-int (0,1), x-int none integer | (−∞,∞) | [0, ∞) |
| y = x^2 + 4 | (0, 4) | x = 0 | Up | x-ints (−2,0),(2,0); y-int (0,4) | (−∞,∞) | [4, ∞) |
| y = (x + 2)^2 − 1 | (−2, −1) | x = −2 | Up | x-ints (−3,0),(1,0); y-int (0,3) | (−∞,∞) | [−1, ∞) |
| y = −2(x − 1)^2 + 3 | (1, 3) | x = 1 | Down | y-int (0,1); x-ints 1 ± √6/2 | (−∞,∞) | (−∞, 3] |
| y = x^2 + 2x − 8 | (−1, 9) | x = −1 | Up | x-ints (−4,0),(2,0); y-int (0,−8) | (−∞,∞) | [9, ∞) |
| y = x^2 + 2x − 3 | (−1, −4) | x = −1 | Up | x-ints (−3,0),(1,0); y-int (0,−3) | (−∞,∞) | [−4, ∞) |
(Note: x-intercepts given when found; some values require algebra.)
Word Problem Example (Projectile Motion)
- Given: ball thrown upward with initial speed 16 m/s from 32 m cliff.
- Height function: h(t) = −4.9t^2 + 16t + 32.
- Time to max height: t = −b/(2a) = 16 / (2·4.9) ≈ 1.633 s.
- Maximum height: h(1.633) ≈ 45.06 m.
- Time to hit ground: solve h(t) = 0 with quadratic formula → positive root ≈ 4.67 s.
- Ignore negative time root for physical interpretation.
Writing Equation From Graph (Given Points)
- If vertex (h,k) and a point (x,y) known:
- Use vertex form y = a(x − h)^2 + k; plug (x,y) to solve for a.
- Convert to standard form by expanding if needed.
- If x-intercepts known:
- Use factored form y = a(x − r1)(x − r2); plug additional point to find a.
- Expand to standard form and complete the square to convert to vertex form.
Action Items / Next Steps
- Practice: graph various quadratics in both forms; use table centered at vertex.
- Practice: complete the square to convert forms and verify vertex.
- Solve word problems: identify which formula to use and interpret only positive time roots.