Final Practice Exam (ungraded) · Weeks 1–15 (Objectives 1–8)

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
What this is: a low-stakes rehearsal for the cumulative final. It mirrors the real exam's blueprint — same coverage across all eight objectives, the same item-type mix, length, and application-skewed difficulty — but is built from fresh item-bank variants and shares none of the live final's questions.
Settings: ungraded (0 points) · unlimited attempts · feedback shown after submission · opens before the exam window so you can prepare.

This is the human-readable practice exam with its vetted answer key and feedback (released after submission). The import-ready Classic QTI 1.2 is in O-practice-final-week-16-qti.xml (generated by a validated Python script — parses with 20 items). The Canvas placement block is at the bottom.

Integrity note for students. Every item here is a fresh variant — new numbers and contexts — with a pre-computed, vetted answer. None of these are the live final questions. Working them builds the skill the final tests, honestly. The paired live exam is L-final-week-16.md.

No tables or calculators are needed — every value is engineered to land on a clean integer or simple expression by hand. (A basic calculator and Desmos are allowed in the room per the syllabus; nothing here requires them. Reminder: no AI on the real Final — use AI only on the prep tutorial.)

---

Blueprint (mirrors the final)

Coverage is proportional to teaching time, matching the real exam: Obj 1 = 2 · Obj 2 = 2 · Obj 3 = 3 · Obj 4 = 3 · Obj 5 = 2 · Obj 6 = 3 · Obj 7 = 3 · Obj 8 = 2. (The actual final items are not listed here — only the shared structure.)

#	Type	Concept	Objective	Week
1	Multiple choice	Simplify a product of powers (exponent rules)	1	1
2	Multiple choice	Order of operations with a squared negative	1	1
3	Multiple choice	Solve a linear equation	2	2
4	Multiple choice	Solve an absolute-value inequality	2	2
5	Multiple choice	Evaluate a function	3	3
6	Multiple choice	Domain of a rational function	3	3
7	Multiple choice	Composition of functions (f∘g)	3	3
8	Multiple choice	Slope through two points	4	4
9	Multiple choice	Write a line in slope-intercept form	4	4
10	Multiple choice	Solve a 2×2 system	4	5
11	Multiple choice	Factor a trinomial	5	6
12	Matching	Expand / factor (special products & GCF)	5	6
13	Multiple choice	Solve a quadratic by factoring	6	7
14	Multiple choice	Vertex from vertex form	6	9
15	Multiple choice	Discriminant → number of real solutions	6	7
16	Multiple choice	Simplify a rational expression	7	11
17	Multiple choice	Evaluate a rational exponent	7	12
18	Multiple choice	Solve a radical equation	7	12
19	Multiple choice	Evaluate a logarithm	8	14
20	Multiple answer	Select all true log/exponential statements	8	14

Objective totals: Obj 1 = 2 · Obj 2 = 2 · Obj 3 = 3 · Obj 4 = 3 · Obj 5 = 2 · Obj 6 = 3 · Obj 7 = 3 · Obj 8 = 2 → 20 items (ungraded; mirrors the 100-point final's emphasis).

---

Questions, key, and feedback

Objective 1 — Simplify with Properties, Order of Operations & Exponents (Week 1)

Q1 (MC). Simplify (4x⁵)(3x⁻³), writing the result with a positive exponent.
- A. 12x⁻¹⁵
- B. 12x² ✅
- C. 7x²
- D. 12x⁸
Feedback: Multiply the coefficients (4 · 3 = 12) and add the exponents (5 + (−3) = 2): 12x². (A multiplies the exponents; C adds the coefficients.)

Q2 (MC). Evaluate −5² + (−3)².
- A. 34
- B. 16
- C. −16 ✅
- D. −34
Feedback: −5² = −25 (the exponent binds tighter than the sign), and (−3)² = +9; −25 + 9 = −16. (Reading −5² as (−5)² = +25 gives the wrong +34.)

Objective 2 — Linear Equations & Inequalities (Week 2)

Q3 (MC). Solve 7x + 4 = 3x + 24.
- A. x = 3
- B. x = 5 ✅
- C. x = 7
- D. x = −5
Feedback: Subtract 3x → 4x + 4 = 24; subtract 4 → 4x = 20; divide → x = 5. Check: 7(5) + 4 = 39 = 3(5) + 24. ✓

Q4 (MC). Solve the inequality |x + 1| ≤ 4. Which interval is the solution set?
- A. x ≤ 3
- B. −5 ≤ x ≤ 3 ✅
- C. x ≤ −5 or x ≥ 3
- D. −3 ≤ x ≤ 5
Feedback: |expr| ≤ c means −c ≤ expr ≤ c: −4 ≤ x + 1 ≤ 4; subtract 1 throughout → −5 ≤ x ≤ 3. (C is the "≥" pattern; "≤" gives a single "and" interval.)

Objective 3 — Functions: Notation, Domain & Operations (Week 3)

Q5 (MC). For g(x) = 2x² − 3, find g(3).
- A. 9
- B. 15 ✅
- C. 33
- D. −15
Feedback: Substitute 3 for x: 2(3)² − 3 = 2(9) − 3 = 18 − 3 = 15. (C squares the whole 2·3 = 6 first; remember exponents act before the coefficient multiplies.)

Q6 (MC). What is the domain of f(x) = (x − 2)/(x + 5)?
- A. All real numbers
- B. All real numbers except x = 2
- C. All real numbers except x = −5 ✅
- D. All real numbers except x = 5
Feedback: Exclude where the denominator is 0: x + 5 = 0 → x = −5. The domain is all reals except x = −5. (B uses the numerator's zero, which is allowed; D drops the sign.)

Q7 (MC). Let f(x) = x + 4 and g(x) = 3x. Find (f∘g)(2).
- A. 6
- B. 18
- C. 10 ✅
- D. 14
Feedback: (f∘g)(2) = f(g(2)). Inside first: g(2) = 3(2) = 6; then f(6) = 6 + 4 = 10. (Do g first, then f — composition is nesting, not multiplying.)

Objective 4 — Linear Functions, Graphs & Systems (Weeks 4–5)

Q8 (MC). Find the slope of the line through (−1, 4) and (3, 12).
- A. −2
- B. 2 ✅
- C. 1/2
- D. 8
Feedback: m = (y₂ − y₁)/(x₂ − x₁) = (12 − 4)/(3 − (−1)) = 8/4 = 2. (Watch the double negative in the run; C flips rise and run.)

Q9 (MC). Write the equation, in slope-intercept form, of the line with slope −3 that passes through (0, 5).
- A. y = 5x − 3
- B. y = −3x + 5 ✅
- C. y = −3x − 5
- D. y = 3x + 5
Feedback: Slope-intercept is y = mx + b; the slope is −3 and (0, 5) is the y-intercept, so b = 5 → y = −3x + 5. (A swaps the slope and intercept; D drops the negative.)

Q10 (MC). Solve the system 3x + y = 10 and x − y = 2. The solution (x, y) is —
- A. (1, 3)
- B. (3, 1) ✅
- C. (2, 4)
- D. (3, −1)
Feedback: Add the equations (the y-terms cancel): 4x = 12 → x = 3; substitute → 3 − y = 2 → y = 1. Solution (3, 1); check 3(3) + 1 = 10. ✓ (A reverses the coordinates.)

Objective 5 — Polynomials & Factoring (Week 6)

Q11 (MC). Factor completely: x² − 3x − 10.
- A. (x − 2)(x + 5)
- B. (x − 5)(x + 2) ✅
- C. (x − 5)(x − 2)
- D. (x + 5)(x + 2)
Feedback: Find two numbers that multiply to −10 and add to −3: that's −5 and +2, so x² − 3x − 10 = (x − 5)(x + 2). Check (FOIL): x² + 2x − 5x − 10 = x² − 3x − 10. ✓ (A's factors add to +3, not −3.)

Q12 (Matching). Match each expression on the left to its equivalent form on the right (expand or factor as needed).
| Expression | Equivalent form |
|---|---|
| (x − 5)(x + 5) | x² − 25 |
| (x + 2)² | x² + 4x + 4 |
| x² − 3x − 10 | (x − 5)(x + 2) |
| 8x³ − 12x² | 4x²(2x − 3) |
Feedback: (x − 5)(x + 5) is a difference of squares → x² − 25 (no middle term). (x + 2)² is a perfect square → x² + 4x + 4 (the middle term is 2·2·x). x² − 3x − 10 factors to (x − 5)(x + 2). 8x³ − 12x² has GCF 4x² → 4x²(2x − 3).

Objective 6 — Quadratics: Equations, Functions & Graphs (Weeks 7, 9)

Q13 (MC). Solve x² − 2x − 15 = 0.
- A. x = 3, 5
- B. x = 5, −3 ✅
- C. x = −5, 3
- D. x = 5, 3
Feedback: Factor: x² − 2x − 15 = (x − 5)(x + 3) = 0; by the zero-product property, x − 5 = 0 or x + 3 = 0 → x = 5 or x = −3. (D drops the negative on the second root.)

Q14 (MC). The graph of f(x) = (x + 2)² − 9 is a parabola. What is its vertex?
- A. (2, −9)
- B. (2, 9)
- C. (−2, −9) ✅
- D. (−2, 9)
Feedback: In vertex form a(x − h)² + k the vertex is (h, k); here (x + 2) = (x − (−2)) gives h = −2, and k = −9, so the vertex is (−2, −9). (A flips the sign of h — the most common vertex error.)

Q15 (MC). Use the discriminant to determine the number of real solutions of x² + 4x + 8 = 0.
- A. Two real solutions
- B. One real solution
- C. No real solutions ✅
- D. Cannot be determined
Feedback: The discriminant is b² − 4ac = 4² − 4(1)(8) = 16 − 32 = −16. A negative discriminant means no real solutions (the parabola never crosses the x-axis). (A positive value would give two; zero would give one.)

Objective 7 — Rational & Radical Expressions and Equations (Weeks 11–12)

Q16 (MC). Simplify (x² − 25)/(x + 5) (for x ≠ −5).
- A. x + 5
- B. x − 5 ✅
- C. x² − 5
- D. −5
Feedback: Factor the numerator as a difference of squares: (x − 5)(x + 5)/(x + 5); cancel the common factor (x + 5) → x − 5 (with x ≠ −5). (A cancels the wrong factor; you can only cancel common factors.)

Q17 (MC). Evaluate 16^(3/4).
- A. 12
- B. 8 ✅
- C. 2
- D. 64
Feedback: A rational exponent means root then power: 16^(3/4) = (⁴√16)³ = 2³ = 8. (A multiplies 16 by 3/4; C stops at the fourth root without cubing.)

Q18 (MC). Solve √(x − 3) = 5.
- A. x = 8
- B. x = 25
- C. x = 28 ✅
- D. x = 22
Feedback: Square both sides: x − 3 = 25 → x = 28. Check: √(28 − 3) = √25 = 5 ✓ (not extraneous). (D forgets to add 3; A squares the 5 incorrectly.)

Objective 8 — Exponential & Logarithmic Functions (Week 14)

Q19 (MC). Evaluate log₃81.
- A. 27
- B. 4 ✅
- C. 3
- D. 243
Feedback: log₃81 asks "3 to what power equals 81?" — since 3⁴ = 81, the answer is 4. (A divides 81 by 3; the log is the exponent, not the quotient.)

Q20 (Multiple answer — select all that apply). Select all statements that are true.
- A. log₃9 = 2 ✅
- B. log₇1 = 0 ✅
- C. 5² = 10
- D. ln 1 = 0 ✅
- E. log₂16 = 8
Feedback: A is true (3² = 9). B is true (anything to the 0 power is 1, so log₇1 = 0). C is false — 5² = 25, not 10. D is true (ln 1 = log_e 1 = 0, since e⁰ = 1). E is false — log₂16 = 4 (2⁴ = 16), not 8. The three true statements are A, B, and D.

---

Answer key (quick reference)

Q	Answer	Q	Answer
1	B (12x²)	11	B ((x − 5)(x + 2))
2	C (−16)	12	(x−5)(x+5)→x²−25 / (x+2)²→x²+4x+4 / x²−3x−10→(x−5)(x+2) / 8x³−12x²→4x²(2x−3)
3	B (x = 5)	13	B (x = 5, −3)
4	B (−5 ≤ x ≤ 3)	14	C (−2, −9)
5	B (15)	15	C (no real solutions)
6	C (except x = −5)	16	B (x − 5)
7	C (10)	17	B (8)
8	B (2)	18	C (x = 28)
9	B (y = −3x + 5)	19	B (4)
10	B (3, 1)	20	A, B, D

---

Quality gate (H5 — self-checked, computer-verified)

• Structure: 20 items mirroring the final's emphasis — coverage Obj 1 = 2 · Obj 2 = 2 · Obj 3 = 3 · Obj 4 = 3 · Obj 5 = 2 · Obj 6 = 3 · Obj 7 = 3 · Obj 8 = 2 matches the live exam's blueprint exactly (ungraded).
• Single-answer integrity: every multiple-choice item (Q1–Q11, Q13–Q19) has exactly one correct option; the matching item (Q12) pairs all four rows one-to-one; the multiple-answer item keys Q20 → A, B, D (C and E left unselected).
• Arithmetic pre-computed and independently re-verified (Python verify_w16.py, sympy): Q1 (4·3)x^(5+(−3)) = 12x²; Q2 −5² = −25, (−3)² = +9, sum −16; Q3 7x+4=3x+24 → x = 5; Q4 |x+1|=4 → x = −5, 3 (interior interval −5 ≤ x ≤ 3); Q5 2(3)²−3 = 15; Q6 x+5 = 0 → exclude −5; Q7 f(g(2)) = f(6) = 10; Q8 (12−4)/(3−(−1)) = 2; Q9 slope −3, intercept 5; Q10 add equations → x = 3, y = 1; Q11/Q12 factor & expand checks (x²−3x−10 = (x−5)(x+2); (x−5)(x+5) = x²−25; (x+2)² = x²+4x+4; 8x³−12x² = 4x²(2x−3)); Q13 (x−5)(x+3) = 0 → x = 5, −3; Q14 vertex (−2, −9); Q15 disc 16−32 = −16 → no real roots; Q16 (x²−25)/(x+5) = x−5; Q17 16^(3/4) = 8; Q18 √(x−3) = 5 → x = 28 (checks); Q19 log₃81 = 4; Q20 log₃9 = 2, log₇1 = 0, ln 1 = 0 (true), 5² = 25 ≠ 10 and log₂16 = 4 ≠ 8 (false). All checks PASS (0 failures).
• Clean values by design: every item resolves to an integer, a simple ordered pair, or a one-term expression — no calculator or table is required, and every distractor encodes a specific, named error.
• QTI parse confirmation: O-practice-final-week-16-qti.xml parses as imsqti_xmlv1p2 with 20 items; every single-answer respcondition sets SCORE = 100 on exactly one option; the multiple-answer item awards 100 only for the exact correct-set selection (A, B, D).
• Integrity vs. the live final: 0 items are shared with L-final-week-16.md (verified by full stem-plus-options comparison; every shared concept slot uses different numbers and contexts).
• No content outside the Weeks 1–15 course definitions; no hallucinated facts.

---

Item-bank & coverage note

All 20 items are fresh variants assembled from the Week 1–15 item banks per Prompt O — drawn so that they mirror the final's blueprint while sharing none of its live items — tagged course=MATH120 · practice=final · weeks=1–15 · objectives=1–8 and deposited back into the banks for future per-term ($39) regenerations. Each term's update regenerates fresh practice variants alongside the live final, and the practice form continues to share none of the live exam's items.

Canvas placement block

canvas_object             = Quizzes::Quiz
title                     = "Final Practice Exam (ungraded)"
assignment_group          = "Practice exercises"
points_possible           = 0
grading_type              = not_graded
allowed_attempts          = unlimited
show_feedback             = true        # released after submission
available_from_offset_days = -5        # opens 5 days before the exam window
due_offset_days           = 4         # on or before the final's due date
published                 = true
shuffle_answers           = true
provenance                = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"