Week 9 — Quiz (auto-graded) · Quadratic Functions & Their Graphs

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective tested: Objective 6 — vertex form, standard form, axis of symmetry, intercepts, direction, minimum/maximum, discriminant.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 9.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-09-qti.xml. AI is not permitted on quizzes (course AI policy). Every numeric answer below is pre-computed and independently re-verified (Python w09_verify.py, PASS — 42 checks, 0 failures).

Blueprint

No trick questions; distractors target the Week 9 misconceptions named in the lecture outline (wrong sign for h in vertex form, dropping the negative in −b/(2a), confusing axis with vertex, confusing minimum with maximum).

Questions, key, and feedback

Q1 (MC). What is the vertex of f(x) = (x − 2)² + 3?
- A. (2, 3) ✅
- B. (−2, 3)
- C. (2, −3)
- D. (−2, −3)
Feedback: Vertex form is f(x) = a(x − h)² + k; the vertex is (h, k). Since the formula has (x − 2), h = 2 and k = 3. (B is the classic sign error: reading (x − 2) as h = −2 instead of h = 2. The subtraction is already built into the formula.)

Q2 (MC). Does f(x) = −2x² + 4x − 1 open upward or downward?
- A. Upward
- B. Downward ✅
- C. Neither — it's a horizontal line
- D. Cannot be determined without graphing
Feedback: The direction is determined solely by the sign of a (the coefficient of x²). Here a = −2 < 0 → the parabola opens downward. (A is the error of ignoring the sign; C and D are nonsense for a quadratic.)

Q3 (MC). Using x = −b/(2a), what is the vertex of f(x) = x² − 6x + 5?
- A. (3, −4) ✅
- B. (−3, −4)
- C. (3, 4)
- D. (6, 5)
Feedback: a = 1, b = −6. Vertex x = −(−6)/(2·1) = 6/2 = 3. Then f(3) = 9 − 18 + 5 = −4. Vertex = (3, −4). (B uses b/(2a) = −6/2 = −3 without the required minus sign; C gets x right but drops the negative from the y-value; D uses the coefficients directly.)

Q4 (MC). What is the axis of symmetry of y = x² − 6x + 5?
- A. y = 3
- B. x = 3 ✅
- C. The point (3, −4)
- D. x = −3
Feedback: The axis of symmetry is a vertical line, always written as x = (number). For this function the vertex x-coordinate is 3, so the axis is x = 3. (A confuses a horizontal line; C confuses the axis with the vertex point; D uses the wrong sign.)

Q5 (MC). What is the y-intercept of y = x² − 6x + 5?
- A. (0, −6)
- B. (0, 1)
- C. (0, 5) ✅
- D. (0, −4)
Feedback: The y-intercept is f(0). For f(x) = ax² + bx + c, f(0) = c. Here c = 5 → y-intercept = (0, 5). (A gives the b coefficient; B invents 1; D gives the vertex y-value.)

Q6 (MC). What are the x-intercepts of y = x² − 6x + 5?
- A. (−1, 0) and (−5, 0)
- B. (0, 1) and (0, 5)
- C. (1, 0) and (5, 0) ✅
- D. (3, 0) only
Feedback: Set x² − 6x + 5 = 0 and factor: (x − 1)(x − 5) = 0 → x = 1 and x = 5 → points (1, 0) and (5, 0). (A gets the wrong-sign roots; B swaps x and y coordinates; D gives only the axis of symmetry.)

Q7 (MC). What is the minimum value of y = x² − 6x + 5?
- A. 3
- B. 5
- C. −4 ✅
- D. The function has no minimum.
Feedback: Since a = 1 > 0, the parabola opens up and the vertex (3, −4) is the minimum. The minimum value (the smallest y-output) is the y-coordinate of the vertex: −4. (A gives the vertex x; B gives the y-intercept; D is wrong — all upward-opening parabolas have a minimum.)

Q8 (MC). What is the maximum value of y = −x² + 4x + 1?
- A. 5 ✅
- B. 2
- C. 1
- D. The function has no maximum.
Feedback: a = −1 < 0 → opens down → the vertex is the maximum. Vertex x = −4/(2·(−1)) = 2; y = −4 + 8 + 1 = 5. Maximum value = 5. (B gives the vertex x; C gives the y-intercept (constant term); D is wrong — downward-opening parabolas always have a maximum.)

Q9 (Matching). Match each vertex-form expression to its vertex.

Feedback: (x − 3)² − 1: h = 3, k = −1 → vertex (3, −1). (x + 1)² = (x − (−1))²: h = −1, k = 4 → vertex (−1, 4). 2x² − 5 = 2(x − 0)² − 5: h = 0, k = −5 → vertex (0, −5). The key in all three: read h as the number subtracted from x inside the square, and don't forget the sign.

Q10 (MC). How many real x-intercepts does y = x² + 1 have?
- A. Two
- B. One
- C. Zero ✅
- D. Cannot be determined
Feedback: Discriminant = b² − 4ac = 0² − 4(1)(1) = −4 < 0. A negative discriminant means no real x-intercepts. The parabola (vertex (0,1), opens up) floats entirely above the x-axis. (A and B would require a non-negative discriminant; D is wrong — the discriminant tells us definitively.)

Answer key (quick reference)

Quality gate (self-checked, computer-verified): each single-answer item has exactly one correct option; the matching item pairs all three expressions one-to-one. Arithmetic pre-computed and independently re-verified (w09_verify.py, PASS — 42 checks, 0 failures): Q1 h=2,k=3 ✓; Q2 a=−2<0 opens down ✓; Q3 −(−6)/2=3, f(3)=−4 ✓; Q4 axis x=3 ✓; Q5 f(0)=5 ✓; Q6 roots 1,5 ✓; Q7 min=−4 ✓; Q8 vertex x=2, max=5 ✓; Q9 three pairs verified ✓; Q10 disc=−4<0 ✓. QTI parse confirmation: F-quiz-week-09-qti.xml — see qtigen output below.

Item-bank entries (for variants + the final)

All ten items are tagged course=MATH120 · week=9 · objective=6 · topic=quadratic-functions-graphs and deposited in Item Bank: Week 9 — Quadratic Functions & Their Graphs. The final (Week 16) and per-term variant updates draw fresh items from this bank. (Tags: q1 vertex-form, q2 opening-direction, q3 vertex-standard, q4 axis-symmetry, q5 y-intercept, q6 x-intercepts, q7 minimum, q8 maximum, q9 vertex-form-matching, q10 discriminant.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 9 Quiz — Quadratic Functions & Their Graphs"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 6        # 6 days after module start (Oct 26 → Nov 1)
published       = true
shuffle_answers = true
provenance      = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"