Overview
Practice test for Algebra/Geometry/Statistics 2, Module 3. Focus: quadratic forms, graph features, and conversions among standard, factored, and vertex forms.
Part I: Multiple Choice Topics
- Identify vertex from vertex form f(x) = (x − h)² + k → vertex (h, k).
- Identify x-intercepts from factored form f(x) = a(x − r1)(x − r2) → intercepts (r1, 0), (r2, 0).
- Factor standard form ax² + bx + c to find factored form.
- Find axis of symmetry from factored form: midpoint of roots x = (r1 + r2)/2.
Quadratic Forms and Key Features
- Standard form: y = ax² + bx + c
- y-intercept at (0, c)
- Axis of symmetry: x = −b/(2a)
- Vertex x-coordinate: −b/(2a); y from substitution
- Factored form: y = a(x − r1)(x − r2)
- x-intercepts: (r1, 0), (r2, 0)
- Axis of symmetry: x = (r1 + r2)/2
- Vertex form: y = a(x − h)² + k
- Vertex: (h, k)
- Axis of symmetry: x = h
Graphing Tasks and Required Outputs
- Given y = (x + 2)(x − 4)
- Find x-intercepts, y-intercept, vertex, axis of symmetry, then graph.
- Given y = (x − 3.5)² + 20.25
- Find x-intercepts, y-intercept, vertex, axis of symmetry, then graph.
- Given y = x² − 5x − 6
- Convert to factored form; find all intercepts, vertex, axis; graph.
- Given y = x² − 6x + 8
- Convert to vertex form; find all intercepts, vertex, axis; graph.
Conversions Practice
- Factored → Standard: expand products, distribute leading coefficients.
- Standard → Factored: factor by grouping or use product-sum for a = 1; factor out GCF first.
- Standard → Vertex: complete the square; for y = ax² + bx + c, convert to a(x − h)² + k.
Structured Practice Items
| Item | Given Form | Task | Target Form/Features |
|---|
| 1 | Vertex form f(x) = (x − 5)² + 3 | Identify vertex | Vertex (5, 3) concept |
| 2 | Factored f(x) = 3(x + 4)(x − 2) | Identify x-intercepts | (−4, 0), (2, 0) concept |
| 3 | Standard f(x) = x² + 5x − 14 | Factor | (x + 7)(x − 2) concept |
| 4 | Factored f(x) = x(x + 7) | Axis of symmetry | x = 3.5 concept |
| 5 | y = (x + 2)(x − 4) | Graph & features | x-ints, y-int, vertex, axis |
| 6 | y = (x − 3.5)² + 20.25 | Graph & features | x-ints, y-int, vertex, axis |
| 7 | y = x² − 5x − 6 | Factor, features, graph | Factored, intercepts, vertex, axis |
| 8 | y = x² − 6x + 8 | Vertex form, features, graph | Vertex form, intercepts, vertex, axis |
| 9 | y = (x − 7)(x + 6) | Expand | Standard form |
| 10 | y = 3(x − 5)(4x + 3) | Expand | Standard form |
| 11 | y = 3x² + 15x + 18 | Factor | Factored form |
| 12 | y = 6x² − x − 2 | Factor | Factored form |
| 13 | y = x² + 10x + 24 | Complete square | Vertex form |
| 14 | y = 3x² − 18x + 110 | Complete square | Vertex form |
Grading Criteria (Learning Targets)
- Communication and organization
- Clear reasoning and well-organized work across Exercises 5–14
- Correct mathematical representations used consistently
- Graph features mastery
- Vertex and axis: Exercises 1, 4, 5–8
- x-intercepts: Exercises 2, 5–8
- y-intercepts: Exercises 5–8
- Conversions proficiency
- Factored → Standard: Exercises 9–10
- Standard → Factored: Exercises 3, 7, 11, 12
- Standard → Vertex: Exercises 8, 13, 14
Key Terms & Definitions
- Vertex: Highest/lowest point of a parabola; from vertex form (h, k)
- Axis of symmetry: Vertical line through vertex; x = h or x = −b/(2a)
- x-intercepts: Points where y = 0; solve factors for x
- y-intercept: Point where x = 0; equals c in standard form
- Completing the square: Method to convert standard to vertex form
Action Items / Next Steps
- Practice factoring quadratics with and without leading coefficient a ≠ 1.
- Drill completing the square to obtain vertex form quickly.
- For each graphing item, list intercepts, vertex, and axis before sketching.