Week 16 · Study guide

Final Exam Study Guide · Weeks 1–15 (Objectives 1–8)

College Algebra · MATH 120 Fall 2026 · Prof. Calloway Fictional sample

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
This is a student-facing review page. Read it, work the fresh practice, and follow the dated plan. Then run the paired Exam-Prep Tutorial and take the Practice Final for active recall. (This guide points to those two — it does not repeat them.)

Integrity note for students. Every practice item on this page 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, the honest way.

What the final covers (read this first)


Exam	Final — cumulative, Weeks 1–15, all 8 Objectives
Format	20 items, 100 points (5 each). Application-skewed: most items ask you to do something — simplify, solve, evaluate, factor, interpret — not just recite a definition. All items are auto-graded (multiple-choice, multiple-answer, matching); no AI on the Final.
Coverage (where the points are)	Obj 1 = 2 items (simplify, exponents, order of operations) · Obj 2 = 2 (linear equations & inequalities, absolute value) · Obj 3 = 3 (functions: evaluate, domain, composition) · Obj 4 = 3 (slope, lines, systems) · Obj 5 = 2 (factoring) · Obj 6 = 3 (quadratics: solve, vertex, discriminant) · Obj 7 = 3 (rational & radical expressions/equations) · Obj 8 = 2 (logarithms). The function-through-logs half (Objectives 3–8) is the heaviest single block (~16 of 20). Budget the most time there.
Weight	The final is 30% of your course grade — the single biggest assessment in the course.
When it opens / where	Opens in the Week 16 module (the final-review-and-exam week). The exam window opens Mon Dec 14 and the exam is due Fri Dec 18, 11:59 p.m. This guide and the exam-prep tutorial post before it so you can prepare. There is no weekly quiz, assignment, or discussion in Week 16 — the final replaces them.
What to bring	A pencil and a basic calculator. Every item is engineered to land on a clean integer or a simple expression — you won't need a table, and the calculator is a backup. Desmos is allowed in the room for checking a graph; nothing on the exam requires it.

How to use this guide. Each objective below has the same four parts: (A) the key ideas in plain language, (B) the definitions / formulas / procedures, (C) the predictable mistakes and their cures, and (D) where to review in the module. After all eight objectives come fresh worked examples + self-check questions (with answers), a dated study plan sized to finals week, and how it's graded + test strategy.

The handful of facts the exam leans on (build these into a one-page sheet):

Exponent rules: xᵃ·xᵇ = xᵃ⁺ᵇ (add) · (xᵃ)ᵇ = xᵃᵇ (multiply) · x⁻ⁿ = 1/xⁿ · x⁰ = 1.

The −4² trap: −4² = −16, but (−4)² = +16.

Absolute value: |expr| = c → expr = ±c · |expr| < c → −c < expr < c · |expr| > c → expr < −c or expr > c.

Slope: m = (y₂ − y₁)/(x₂ − x₁) (rise over run) · slope-intercept y = mx + b.

Special products: (a − b)(a + b) = a² − b² · (a ± b)² = a² ± 2ab + b².

Vertex form: a(x − h)² + k has vertex (h, k) (the x-coordinate is +h).

Discriminant: b² − 4ac > 0 → two real · = 0 → one · < 0 → none.

Rational exponent: a^(m/n) = (ⁿ√a)ᵐ (bottom is the root, top is the power).

Logarithm: logb x = y ↔ bʸ = x · logb 1 = 0 · logb b = 1 · ln e = 1.

Objective 1 — Simplify with Properties, Order of Operations & Exponents (Week 1) · 2 items

(A) Key ideas, plain language

To simplify is to rewrite an expression in a cleaner, equal form — without changing its value. Two things cost the most points here: the order of operations and the rules for exponents. Get the signs right and the exponent rules straight and Objective 1 is free points.

(B) Definitions, formulas, procedures

Real-number sets: natural ⊂ whole ⊂ integer ⊂ rational ⊂ real; the irrationals (√2, π) fill the gaps. A radical is irrational only when it doesn't simplify (√16 = 4 is rational).
Order of operations (PEMDAS as two left-to-right pairs): Parentheses → Exponents → (Multiply/Divide, ties, left to right) → (Add/Subtract, ties, left to right).
The sign trap: −4² = −16 (only the 4 is squared); (−4)² = +16 (the parentheses square the whole −4).
Integer-exponent rules: xᵃ·xᵇ = xᵃ⁺ᵇ (add); (xᵃ)ᵇ = xᵃᵇ (multiply); xᵃ/xᵇ = xᵃ⁻ᵇ; x⁻ⁿ = 1/xⁿ; x⁰ = 1 (x ≠ 0); (xy)ⁿ = xⁿyⁿ.
Distributive property: a(b + c) = ab + ac — distribute to every term, watching the sign: −(x − 5) = −x + 5.

(C) Predictable mistakes → cures

❌ "−4² = 16." → ✅ The exponent binds tighter than the sign → −16. You get +16 only inside parentheses.
❌ "x²·x³ = x⁶" (multiplying exponents). → ✅ Same base times same base → add → x⁵. Multiply only for a power of a power.
❌ "x⁻² means negative." → ✅ A negative exponent means reciprocal: x⁻² = 1/x².
❌ Drops a sign distributing a negative. → ✅ −(x − 5) = −x + 5; the negative hits every term.
❌ "√20 is rational." → ✅ √20 = 2√5 (doesn't reduce to a whole number/fraction) → irrational.

(D) Review in the module

Week 1 → Lecture Outline (order of operations, exponent rules, simplifying), Slides (Deck 1), Readings, Lecture Tutorial 1.

Objective 2 — Linear Equations & Inequalities (Week 2) · 2 items

(A) Key ideas, plain language

To solve is to find the value(s) that make the statement true, by doing the same operation to both sides until the variable is alone. Inequalities work the same way — with one extra rule — and absolute value splits into cases.

(B) Definitions, formulas, procedures

Linear equation: isolate x by undoing operations in reverse order. Always check by substituting back.
Inequality: solve like an equation, but flip the inequality sign when you multiply or divide by a negative. Express the answer in interval notation if asked: x > 3 ↔ (3, ∞); −2 ≤ x < 5 ↔ [−2, 5).
Absolute-value equation: |expr| = c (c ≥ 0) → expr = c or expr = −c.
Absolute-value inequality:
|expr| < c → −c < expr < c (a single "and" interval — between).
|expr| > c → expr < −c or expr > c (two pieces — outside).

(C) Predictable mistakes → cures

❌ Forgets to flip the sign when dividing an inequality by a negative. → ✅ −2x < 6 → x > −3 (sign flips).
❌ Treats |expr| < c like an "or." → ✅ "Less than" is between: −c < expr < c. "Greater than" is the or.
❌ Writes only one solution to |x| = 5. → ✅ Two: x = 5 and x = −5.
❌ Drops a sign moving a term across. → ✅ Move terms by doing the inverse to both sides; recheck the sign.

(D) Review in the module

Week 2 → Lecture Outline (linear equations, inequalities & interval notation, absolute value), Slides (Deck 2), Lecture Tutorial 2.

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

(A) Key ideas, plain language

A function is a rule that takes one input and returns exactly one output. f(x) is not multiplication — it's "the output of f at x." Three moves: evaluate, find the domain, and compose.

(B) Definitions, formulas, procedures

Evaluate f(a): substitute a for every x and simplify. Mind the signs: f(−2) with f(x) = x² + 1 is (−2)² + 1 = 5 (the square makes it positive).
Domain = all legal inputs. Exclude two things: a zero denominator (set any denominator ≠ 0) and a negative under an even root (set the inside ≥ 0). For (x + 1)/(x − 3), exclude x = 3.
Operations on functions: (f + g)(x), (f − g)(x), (fg)(x), (f/g)(x) — combine the rules termwise (and exclude where g = 0 for the quotient).
Composition: (f∘g)(x) = f(g(x)) — do the inside (g) first, then feed it to f. Order matters: (f∘g) ≠ (g∘f) in general.

(C) Predictable mistakes → cures

❌ "f(−2) = f times −2." → ✅ f is a machine — substitute −2 for x.
❌ Forgets parentheses on a negative input. → ✅ f(−2) with x² → (−2)² = +4, not −4.
❌ Gives a domain without excluding anything. → ✅ Ask what's illegal? — zero denominator or negative even-root.
❌ Composes by multiplying, or in the wrong order. → ✅ (f∘g)(x) = f(g(x)): inside first, then outside.

(D) Review in the module

Week 3 → Lecture Outline (function notation & evaluation, domain & range, operations & composition), Slides (Deck 3), Lecture Tutorial 3.

Objective 4 — Linear Functions, Graphs & Systems (Weeks 4–5) · 3 items

(A) Key ideas, plain language

A line is the graph of a constant rate of change. Find its slope, write it in slope-intercept form, and where two lines meet is the solution of a system.

(B) Definitions, formulas, procedures

Slope between (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁) = rise over run. Sign = direction; size = steepness.
Slope-intercept form: y = mx + b (m = slope, b = y-intercept). Point-slope form: y − y₁ = m(x − x₁).
Parallel/perpendicular: parallel → same slope; perpendicular → negative reciprocal slopes (2 and −1/2).
Systems (2×2): the solution is the point (x, y) satisfying both equations.
Substitution: solve one equation for a variable, plug into the other.
Elimination: add/subtract the equations to cancel a variable.
Special cases: no solution (parallel lines), infinitely many (same line).

(C) Predictable mistakes → cures

❌ Slope as run over rise. → ✅ Rise over run: change in y over change in x.
❌ Mishandles minus signs in the differences. → ✅ (8 − (−1))/(5 − 2) = 9/3 = 3 — the double negative is a +.
❌ Stops a system at one number. → ✅ The answer is an ordered pair (x, y) — solve for both.
❌ Forgets to check in BOTH equations. → ✅ A correct (x, y) must satisfy each line.

(D) Review in the module

Week 4 → Lecture Outline (slope, slope-intercept & point-slope, graphing, parallel/perpendicular), Deck 4, Tutorial 4. Week 5 → Lecture Outline (substitution, elimination, consistent/inconsistent/dependent), Deck 5, Tutorial 5.

Objective 5 — Polynomials & Factoring (Week 6) · 2 items

(A) Key ideas, plain language

Factoring is multiplication run backwards — rewrite a sum as a product. It's the key that unlocks quadratics (Objective 6) and simplifies rational expressions (Objective 7). The order of attack always starts with the GCF.

(B) Definitions, formulas, procedures

GCF first, always: pull the greatest common factor out of every term. 6x³ − 9x² = 3x²(2x − 3).
Trinomial x² + bx + c: find two numbers that multiply to c and add to b → (x + ?)(x + ?). x² − 5x − 14: −7 and +2 → (x − 7)(x + 2).
Difference of squares: a² − b² = (a − b)(a + b) — two perfect squares with a minus, no middle term. x² − 16 = (x − 4)(x + 4). A sum of squares (x² + 16) does NOT factor over the reals.
Perfect-square trinomial: a² + 2ab + b² = (a + b)² (and a² − 2ab + b² = (a − b)²). x² + 6x + 9 = (x + 3)².
Special products to expand: (a + b)(a − b) = a² − b²; (a ± b)² = a² ± 2ab + b².

(C) Predictable mistakes → cures

❌ Skips the GCF. → ✅ Always pull the GCF first — it exposes the structure.
❌ Factors a SUM of squares. → ✅ x² + 16 is prime over the reals; only the difference factors.
❌ Sign errors in the trinomial pair. → ✅ Check by expanding (FOIL): (x − 7)(x + 2) = x² − 5x − 14. ✓
❌ Calls a trinomial a difference of squares. → ✅ A difference of squares has no middle term.

(D) Review in the module

Week 6 → Lecture Outline (polynomial operations, special products, GCF/trinomials/difference of squares/grouping), Deck 6, Tutorial 6.

Objective 6 — Quadratics: Equations, Functions & Graphs (Weeks 7, 9) · 3 items

(A) Key ideas, plain language

A quadratic is degree two; its graph is a parabola. Solve it (factoring or the formula), read its vertex, and use the discriminant to know how many real solutions exist before solving.

(B) Definitions, formulas, procedures

Zero-product property: if a product = 0, at least one factor is 0. Factor, set each factor to 0, solve. x² − 4x − 5 = (x − 5)(x + 1) = 0 → x = 5 or x = −1.
Quadratic formula (when it won't factor): x = (−b ± √(b² − 4ac)) / (2a).
Vertex form: f(x) = a(x − h)² + k has vertex (h, k) — the x-coordinate is +h (the minus is already in the form). f(x) = (x − 1)² − 4 → vertex (1, −4).
Axis of symmetry: the vertical line x = h. For y = ax² + bx + c, h = −b/(2a).
Discriminant b² − 4ac: > 0 → two real solutions · = 0 → one · < 0 → none (the parabola misses the x-axis).

(C) Predictable mistakes → cures

❌ Gives only one root from (x − 5)(x + 1) = 0. → ✅ Every factor gives a root: x = 5 and x = −1.
❌ Vertex of (x − 1)² − 4 is (−1, −4). → ✅ The x-coordinate is +h = +1; the form's minus is built in.
❌ Forces a real answer when the discriminant is negative. → ✅ b² − 4ac < 0 → no real solutions.
❌ Drops a negative root. → ✅ x = 5 and −1 — keep the sign on the second solution.

(D) Review in the module

Week 7 → Lecture Outline (factoring & zero-product, completing the square, quadratic formula & discriminant), Deck 7, Tutorial 7. Week 9 → Lecture Outline (parabolas & vertex, vertex/standard forms, intercepts, axis of symmetry, max/min), Deck 9, Tutorial 9.

Objective 7 — Rational & Radical Expressions and Equations (Weeks 11–12) · 3 items

(A) Key ideas, plain language

Objective 7 extends the toolkit to fractions with variables and roots. Simplify a rational expression (factor, then cancel — and note the restriction), evaluate a rational exponent, and solve a radical equation (always checking for extraneous answers).

(B) Definitions, formulas, procedures

Simplify a rational expression: factor top and bottom, cancel common factors, and keep the restriction (the original denominator's zero is still excluded). (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3, x ≠ 3.
You can only cancel FACTORS, not terms. (x² − 9)/(x − 3) ≠ x − 3.
Rational exponents: a^(m/n) = (ⁿ√a)ᵐ — bottom is the root, top is the power. 27^(2/3) = (³√27)² = 3² = 9. Also a^(1/n) = ⁿ√a.
Solve a radical equation: isolate the radical, raise both sides to the matching power, solve, then check for extraneous solutions (squaring can create false answers). √(x + 1) = 4 → x + 1 = 16 → x = 15 (check √16 = 4 ✓).

(C) Predictable mistakes → cures

❌ Cancels loose terms. → ✅ Factor first, then cancel the common factor (x − 3) → x + 3.
❌ "27^(2/3) = 18." → ✅ Bottom is the root: (³√27)² = 9.
❌ Skips the extraneous check. → ✅ Substitute back into the original radical equation, every time.
❌ Forgets the restriction on a simplified rational expression. → ✅ Carry x ≠ 3 along (the value that broke the original).

(D) Review in the module

Week 11 → Lecture Outline (simplifying rational expressions, operations, solving rational equations & extraneous solutions), Deck 11, Tutorial 11. Week 12 → Lecture Outline (simplifying radicals, rational exponents, solving radical equations & extraneous solutions), Deck 12, Tutorial 12.

Objective 8 — Exponential & Logarithmic Functions (Week 14) · 2 items

(A) Key ideas, plain language

An exponential function (bˣ, or eˣ) grows or shrinks by a constant factor. A logarithm is its inverse — it answers "what exponent?" Master the inverse relationship and every log value becomes obvious.

(B) Definitions, formulas, procedures

The inverse relationship (the whole key): logb(x) = y ↔ bʸ = x. ("log base b of x is the power you raise b to, to get x.")
Evaluate a log by asking the exponent question: log₂16 = "2 to what power = 16?" → 4 (2⁴ = 16).
Key values: logb 1 = 0 (b⁰ = 1, any base) · logb b = 1 · ln e = 1 · log₁₀100 = 2 (10² = 100).
Natural log: ln x = log_e x, base e ≈ 2.718. Common log: log x = log₁₀ x.
Domain of a log: the argument must be positive (logb of 0 or a negative is undefined).
(For solving — Week 15 — you use the same-base method or take a log of both sides; the Final focuses on evaluating and the inverse relationship.)

(C) Predictable mistakes → cures

❌ "log₂8 = 4." → ✅ It's 3 (2³ = 8) — the log is the exponent.
❌ "log₅1 = 5" or "= 1." → ✅ logb 1 = 0 for every base (b⁰ = 1).
❌ "ln e = 0." → ✅ ln e = 1 (e¹ = e).
❌ "log₁₀100 = 10." → ✅ It's 2 (10² = 100) — ask "10 to what power is 100?"
❌ Divides instead of asking the exponent question. → ✅ log₂16 ≠ 16/2; it's the power (4).

(D) Review in the module

Week 13 → Lecture Outline (exponential functions & graphs, growth/decay, base e), Deck 13, Tutorial 13. Week 14 → Lecture Outline (logs as inverses, graphs & domain, properties of logarithms), Deck 14, Tutorial 14.

Representative practice (all fresh — vetted answers)

None of these are live final items. New numbers, new contexts. Each answer is pre-computed; the one-line why names the idea it tests. Cover the answers, work each one, then check. Practice is weighted toward the heavier objectives (3–8).

Objective 1 practice

Worked example — simplify with exponents and the sign trap.
- (a) Simplify (2x³)(5x⁴). (b) Simplify (x⁵)/(x²). (c) Evaluate −3² + (−2)⁴. (d) Rewrite 4x⁻³ with a positive exponent.
Answer. (a) 2·5 = 10, add exponents 3 + 4 = 7 → 10x⁷. (b) subtract exponents 5 − 2 = 3 → x³. (c) −3² = −9 and (−2)⁴ = +16 → −9 + 16 = 7. (d) 4/x³. Why: multiply powers → add exponents; the exponent binds tighter than the sign; a negative exponent is a reciprocal.

Self-check (Obj 1).
1. Is √36 rational or irrational? → Rational (= 6).
2. Simplify (x³)². → x⁶ (power of a power → multiply).
3. Evaluate (−5)². → 25 (parentheses square the whole −5).
4. Distribute −(2x − 7). → −2x + 7.

Objective 2 practice

Worked example — linear equation + absolute-value inequality.
- (a) Solve 4x + 5 = 2x + 17. (b) Solve −3x > 12. (c) Solve |x − 1| < 3 and give the interval.
Answer. (a) subtract 2x → 2x + 5 = 17; subtract 5 → 2x = 12 → x = 6. (b) divide by −3 and flip: x < −4. (c) −3 < x − 1 < 3; add 1 → −2 < x < 4. Why: same operation to both sides; flip the sign when dividing by a negative; |expr| < c is a between-interval.

Self-check (Obj 2).
1. Solve 2(x − 3) = 10. → x = 8.
2. |x| = 7 has how many solutions? → Two (x = 7 and −7).
3. Solve |x + 2| > 5 (form of the answer). → x < −7 or x > 3.
4. Solve x/(−2) ≤ 3. → x ≥ −6 (flip the sign).

Objective 3 practice

Worked example — evaluate, domain, composition.
Let f(x) = x² − 2x and g(x) = x + 3.
- (a) f(4). (b) Domain of h(x) = (x − 1)/(x + 2). (c) (f∘g)(0) — actually compute (g∘f)(2) too.
Answer. (a) f(4) = 16 − 8 = 8. (b) exclude x + 2 = 0 → all reals except x = −2. (c) (f∘g)(0) = f(g(0)) = f(3) = 9 − 6 = 3; (g∘f)(2) = g(f(2)) = g(0) = 3 (here they coincide, but in general order matters). Why: substitute for every x; a domain excludes a zero denominator; composition is inside-first nesting.

Self-check (Obj 3).
1. For f(x) = 3x − 1, find f(−2). → −7.
2. Domain of √(x − 4)? → x ≥ 4.
3. (f∘g)(x) means? → f(g(x)) — g first.
4. Is f(x) = x² a function? → Yes (each input has one output).

Objective 4 practice

Worked example — slope, line, system.
- (a) Slope through (1, 2) and (4, 14). (b) Line with slope −2 through (0, 7). (c) Solve x + y = 8, x − y = 2.
Answer. (a) (14 − 2)/(4 − 1) = 12/3 = 4. (b) y = −2x + 7. (c) add the equations → 2x = 10 → x = 5; then 5 + y = 8 → y = 3 → (5, 3). Check x − y = 2 ✓. Why: slope is rise over run; slope-intercept reads off m and b; a system's answer is a point.

Self-check (Obj 4).
1. Slope of a horizontal line? → 0.
2. Perpendicular to slope 3? → −1/3.
3. Solve y = 2x, x + y = 9. → (3, 6).
4. y-intercept of y = 4x − 5? → −5 (the point (0, −5)).

Objective 5 practice

Worked example — factor four ways.
- (a) GCF: 10x⁴ − 15x³. (b) Trinomial: x² + 7x + 12. (c) Difference of squares: x² − 49. (d) Perfect square: x² − 8x + 16.
Answer. (a) GCF 5x³ → 5x³(2x − 3). (b) 3 and 4 multiply to 12, add to 7 → (x + 3)(x + 4). (c) (x − 7)(x + 7). (d) (x − 4)² (since 2·4 = 8). Why: GCF first; trinomial pair multiplies to c, adds to b; difference of squares; perfect-square pattern.

Self-check (Obj 5).
1. Factor x² − x − 6. → (x − 3)(x + 2).
2. Does x² + 9 factor over the reals? → No (sum of squares).
3. Expand (x − 5)². → x² − 10x + 25.
4. GCF of 12x² + 8x? → 4x → 4x(3x + 2).

Objective 6 practice

Worked example — solve, vertex, discriminant.
- (a) Solve x² + x − 6 = 0. (b) Vertex of f(x) = (x − 3)² + 2. (c) Discriminant and # real solutions of x² − 6x + 9 = 0.
Answer. (a) (x + 3)(x − 2) = 0 → x = −3 or x = 2. (b) vertex form → (3, 2). (c) b² − 4ac = 36 − 36 = 0 → one real solution (x = 3, a double root). Why: zero-product gives both roots; vertex x is +h; discriminant 0 → exactly one real solution.

Self-check (Obj 6).
1. Solve (x − 4)(x + 1) = 0. → x = 4, −1.
2. Vertex of (x + 5)² − 1? → (−5, −1).
3. Discriminant of x² + x + 1? → 1 − 4 = −3 → no real solutions.
4. Axis of symmetry of y = (x − 2)² + 3? → x = 2.

Objective 7 practice

Worked example — rational simplify, rational exponent, radical equation.
- (a) Simplify (x² − 4)/(x + 2). (b) Evaluate 8^(2/3). (c) Solve √(x + 5) = 6.
Answer. (a) (x − 2)(x + 2)/(x + 2) = x − 2, x ≠ −2. (b) (³√8)² = 2² = 4. (c) square → x + 5 = 36 → x = 31; check √36 = 6 ✓. Why: factor before cancelling and keep the restriction; bottom-is-the-root; isolate, square, and check for extraneous.

Self-check (Obj 7).
1. Simplify (x² − 1)/(x − 1). → x + 1, x ≠ 1.
2. Evaluate 9^(1/2). → 3.
3. Solve √x = 7. → x = 49 (checks).
4. Evaluate 16^(1/4). → 2.

Objective 8 practice

Worked example — evaluate logs via the inverse relationship.
- (a) log₂32. (b) log₁₀1000. (c) log₄1. (d) Rewrite 2⁵ = 32 in log form.
Answer. (a) "2 to what power = 32?" → 5 (2⁵ = 32). (b) "10 to what power = 1000?" → 3. (c) 0 (any base of 1 is 0). (d) log₂32 = 5. Why: a log is the exponent; logb 1 = 0; exponential and log forms are the same statement.

Self-check (Obj 8).
1. log₃27 = ? → 3 (3³ = 27).
2. ln 1 = ? → 0.
3. log₅5 = ? → 1.
4. log₂(1/2) = ? → −1 (2⁻¹ = 1/2).

Study plan — a dated countdown (finals week, sized to 2 sessions/week)

Built for the Week 16 final (window opens Mon Dec 14, due Fri Dec 18). Adjust the exact days to your section's posted exam window; the rhythm is what matters. The final is cumulative and the function-through-logs half (Obj 3–8) is the heaviest — start there once your foundations are warm. Do a little every day rather than one long cram.

When	Do this (≈60–90 min)
~7 days out (end of Week 15)	Read this guide's Objectives 1–2 (simplify & solve — the foundation). Work the Obj 1 & 2 practice. Build your one-page formula sheet (exponent rules, the −4² trap, absolute-value cases, slope, special products, vertex form, discriminant, rational-exponent rule, log facts).
~6 days out	Read Objective 3 (evaluate, domain, composition) and Objective 4 (slope, lines, systems). Work both practice sets. Re-derive any you missed.
~5 days out	Read Objective 5 (factoring — GCF first, trinomials, difference of squares, perfect squares). Work its practice until factoring is automatic — it powers Obj 6 and 7.
~4 days out	Read Objective 6 (quadratics: zero-product, vertex form, discriminant). Work its practice. Drill the vertex sign and the "keep both roots."
~3 days out	Read Objectives 7 & 8 (rational & radical expressions/equations; logs via the inverse relationship). Work their practice — nail factor-before-cancel, the extraneous check, bottom-is-the-root, and the log = exponent idea. Then run the paired Exam-Prep Tutorial (`N-exam-prep-tutorial-week-16`) in an approved chatbot — it diagnoses your weak spots across all 8 objectives and drills them.
~2 days out	Take the Practice Final (`O-practice-final-week-16`, the paired practice exam in this module) under timed, closed-note conditions. Score it; list every missed idea by objective.
~1 day out	Re-teach only the topics you missed on the practice final (use this guide's mistake-cures and the relevant Lecture Tutorial). Re-do those specific self-checks, with extra attention to Obj 3–8. Sleep.
Exam day (~Dec 14–18)	Skim your one-page formula sheet. Arrive early. Read each item twice; for every solve/factor item, do the check (substitute back, expand the factors, confirm the radical).

Two paired tools — use both (don't skip):
- Exam-Prep Tutorial (N-exam-prep-tutorial-week-16) — a copy/paste chatbot tutor that diagnoses, re-teaches, and drills you across all 8 objectives, ending with a readiness summary. Best for active recall and shoring up weak spots. (Use AI here, not on the Final.)
- Practice Final (O-practice-final-week-16, the paired practice exam in the Week 16 module) — a full, fresh, timed run that mirrors the real format and the 20-item emphasis. Best for pacing and a final readiness check.

(This guide points to both on purpose — it doesn't duplicate them.)

How the final is graded + test-taking strategy

How it's graded.
- 100 points across 20 items (5 each), weighted toward application (doing, not reciting). Every item is auto-graded (multiple-choice, multiple-answer, matching) — there's no partial credit, so the check at the end of each problem is your safety net.
- The final is 30% of your course grade — the largest single assessment. It replaces Week 16's quiz, assignment, and discussion (there are none that week). No AI on the Final.
- Coverage matches this guide: 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 function-through-logs block (Obj 3–8) is ~16 of 20 — practice it until the procedures are automatic.

Honest test-taking strategies for this material.
1. Respect the signs first. Before anything else, watch −a² vs (−a)², and distribute negatives to every term. The most common lost point in the course is a dropped or mishandled negative.
2. Get the exponent rule right: times → add, power of a power → multiply, negative → reciprocal. Say it before you simplify.
3. For an absolute-value inequality, name the shape: < → between (one "and" interval); > → outside (an "or").
4. For a function, ask "what's illegal?" to find a domain — a zero denominator or a negative under an even root.
5. Slope is rise over run, and a system's answer is an ordered pair (x, y) — never stop at one number.
6. Factor the GCF first, and remember a sum of squares doesn't factor. Check every factoring by expanding.
7. Quadratics: keep BOTH roots, read the vertex x as +h, and trust a negative discriminant (no real solutions).
8. Rational/radical: factor before you cancel, use bottom-is-the-root for rational exponents, and check every radical answer for extraneous solutions.
9. A log is the exponent. Turn logb x = y into bʸ = x and the value is obvious; remember logb 1 = 0 and ln e = 1.
10. Do the easy items first, flag the hard ones, and budget your time — 20 items means a few minutes each. Read each item twice and answer the question actually asked.

Canvas placement block

canvas_object   = Page
title           = "Final Exam Study Guide — Weeks 1–15 (Objectives 1–8)"
module          = "Week 16 — Final Review & Exam"
grading_type    = not_graded
available_from  = 2026-12-07      # posts before the Week 16 final exam window opens
published       = true
provenance      = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"

Term-update note: each term's $39 update regenerates fresh practice variants from this same scope — the live final is never reproduced here.

The per-term $39 update (fresh assessment variants, re-paced to your next calendar) referenced above is on the roadmap — coming soon. Today's download is yours to keep, but it doesn't refresh itself.

~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com