Week 16 — Lecture Outline · Final Review & Exam

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objectives covered: cumulative — Objectives 1–8 (Weeks 1–15). Obj 1 — simplify with real-number properties, order of operations & integer-exponent rules; Obj 2 — solve linear equations & inequalities, incl. absolute value; Obj 3 — functions: notation, domain & range, operations & composition; Obj 4 — linear functions, graphs & systems; Obj 5 — polynomials & factoring; Obj 6 — quadratics: equations, functions & graphs; Obj 7 — polynomial/rational functions and rational & radical expressions/equations; Obj 8 — exponential & logarithmic functions, equations & applications.
SLOs touched: A (apply algebraic procedures accurately) · B (connect symbolic/numerical/graphical representations; interpret in context)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.

This is the final review-and-exam week — no new content. It is cumulative over the entire course (Weeks 1–15, Objectives 1–8). Each segment briskly re-teaches one or two objectives with a quick worked example and the single misconception most likely to cost points; the final segment frames the comprehensive Final and how to prepare. Built to be taught from cold as a capstone review: an instructor (or a substitute) can run it without having taught the course, because every definition, number, and cure travels with the segment. This week's only graded item is the Final (30%) — there is no quiz, no discussion, and no assignment; the Final stands in for all of them. The Final pairs with a Study Guide + Exam-Prep Tutorial + Practice Final, built separately and referenced here by name.

---

Week at a Glance

The week's big question	"Across the whole course — simplifying, solving, graphing, factoring, and modeling — what is the one honest move each topic asks of us, and where does everyone slip?"
By the end of the week, students can…	(1) re-run each objective's core move on demand — simplify with the exponent rules and respect the signs (Obj 1); solve a linear equation or an absolute-value inequality (Obj 2); evaluate, find a domain, and compose functions (Obj 3); find a slope, write a line, and solve a 2×2 system (Obj 4); factor a GCF, a trinomial, and a difference of squares (Obj 5); solve a quadratic, read a vertex, and use the discriminant (Obj 6); simplify a rational expression, evaluate a rational exponent, and solve a radical equation with the extraneous check (Obj 7); evaluate a logarithm and use the exponential↔log inverse relationship (Obj 8); (2) name and avoid the highest-cost misconception in each theme; (3) walk into the Final knowing its coverage, its weight (30%), and a concrete plan built around the Study Guide, the Exam-Prep Tutorial, and the Practice Final.
Key vocabulary (all review)	real-number sets, order of operations (PEMDAS), exponent rules (product/quotient/power, zero, negative), distributive property; linear equation, inequality, interval notation, absolute-value equation/inequality; function notation, domain & range, composition (f∘g); slope, slope-intercept & point-slope form, system of equations (substitution/elimination); GCF, trinomial, difference of squares, perfect-square trinomial; zero-product property, vertex form, axis of symmetry, discriminant; rational expression & restriction, rational exponent, radical equation, extraneous solution; exponential function, base e, logarithm, inverse relationship, growth/decay
Materials	slides (Deck 16 — the final-review deck), the Study Guide, the Exam-Prep Tutorial (AI), the Practice Final, a calculator, Desmos (or GeoGebra) for graphical confirmation, one approved chatbot (Gemini / Claude / ChatGPT) for the audit-the-AI review moment
Timing note	8 segments, ~150 min total. Session 1 (Tue) = Segments 1–4 (~75): the map + Objectives 1–4 (simplify → solve → functions → lines & systems). Session 2 (Thu) = Segments 5–8 (~75): Objectives 5–8 (factoring → quadratics → rational/radical → exponential/log) + the Final frame. Scale to your own pattern.

---

Segment 1 — Hook & the Map of the Whole Course (10 min) · Session 1 opens

Hook. Put one messy expression on the board with no instructions: "−4² + 2(3x − 4) = 5x − 1." Ask the room: "Before you touch this — what skills does it actually take to handle a line like this honestly?" Let them call things out. Steer the harvest: Simplify the constant (and don't get −4² wrong). Distribute the 2 without dropping a sign. Collect like terms. Solve for x. Check. "Every one of those is an objective of this course. Sixteen weeks, eight objectives, and they line up into a single arc — simplify, solve, build functions, graph them, factor, solve quadratics, handle rational and radical pieces, and model growth. Today we walk the whole arc once, fast, and find the exact spot in each chapter where points get lost. That's the Final."

The promise (write it on the board): "By Thursday you'll be able to take any of the eight skills — simplify, solve, evaluate a function, graph a line, factor, solve a quadratic, simplify a rational/radical, evaluate a log — and on demand state the one honest move it requires and the one mistake that sinks it."

The map (one slide, say it out loud — this is the photograph slide of the week):

THE LANGUAGE: Obj 1 simplify expressions · Obj 2 solve equations & inequalities.
THE OBJECTS: Obj 3 functions · Obj 4 lines & systems · Obj 5 polynomials & factoring · Obj 6 quadratics.
THE TOOLKIT GROWS: Obj 7 rational & radical expressions/equations · Obj 8 exponential & logarithmic models.

Why it matters line (memory hook): "The whole course is one arc — first you learn to rewrite without changing the value, then to solve, then to build and graph the objects, and finally to model how the real world grows and shrinks."

---

Segment 2 — Objectives 1 & 2 Review: Simplify, then Solve (20 min)

Re-teach Obj 1 in plain language. To simplify is to rewrite an expression in a cleaner form without changing its value. Three tools carry the unit: the order of operations (PEMDAS as two left-to-right pairs — multiply/divide are ties, add/subtract are ties), the integer-exponent rules (multiply powers → add exponents; power of a power → multiply; xⁿ in a denominator → x⁻ⁿ), and the distributive property (distribute to every term, watching the sign).

Re-teach Obj 2 in plain language. To solve is to find the value(s) that make a statement true, by doing the same thing to both sides until the variable is alone. For inequalities, the one extra rule: flip the inequality sign when you multiply or divide by a negative. For absolute value, split into cases: |expr| = c → expr = c or expr = −c, and |expr| < c → −c < expr < c (a single "and" interval).

One quick worked example (do every step out loud):

Simplify (3x⁴)(2x⁻²), then solve 5x − 3 = 2x + 9.
- Simplify: multiply coefficients (3·2 = 6), add the exponents (4 + (−2) = 2) → 6x². (Multiplying the exponents to get x⁻⁸ is the classic slip.)
- Solve: 5x − 3 = 2x + 9 → subtract 2x → 3x − 3 = 9 → add 3 → 3x = 12 → x = 4. Check: 5(4) − 3 = 17 and 2(4) + 9 = 17. ✓

Highest-cost misconception + cure:
- ❌ "−4² = 16," and "multiplying powers means multiply the exponents."
✅ Cure: −4² = −16 — the exponent binds tighter than the sign, so only the 4 is squared; you get +16 only inside parentheses, (−4)². And xᵃ·xᵇ = xᵃ⁺ᵇ (add); you multiply exponents only for a power of a power, (xᵃ)ᵇ = xᵃᵇ. "Same base times same base → add; power of a power → multiply." (And on inequalities, the most expensive miss is forgetting to flip when dividing by a negative.)

---

Segment 3 — Objective 3 Review: Functions (16 min)

Hook back in: "We can simplify and solve. Now the central object of the whole course — the function — a rule that takes one input and returns exactly one output."

Re-teach in plain language. f(x) is not multiplication — it's "the output of f at the input x." Three moves carry the objective:
- Evaluate: substitute the input everywhere x appears and simplify. f(−2) means "put −2 in for x."
- Domain: the set of legal inputs. Two things to exclude — you can't divide by zero (so set any denominator ≠ 0) and you can't take an even root of a negative (so a square root's inside must be ≥ 0).
- Compose: (f∘g)(x) = f(g(x)) — do the inside (g) first, then feed the result to f. The order matters.

One quick worked example (compute it live):

Let f(x) = x² + 1, g(x) = 2x. Find f(−2), the domain of (x + 1)/(x − 3), and (f∘g)(3) with f(x) = x − 1.
- f(−2) = (−2)² + 1 = 4 + 1 = 5. (Note (−2)² = +4 — the parentheses matter.)
- Domain of (x + 1)/(x − 3): denominator zero at x = 3, so the domain is all reals except x = 3.
- (f∘g)(3) with f(x) = x − 1, g(x) = 2x: inside first, g(3) = 6; then f(6) = 6 − 1 = 5.

Highest-cost misconception + cure:
- ❌ "f(−2) means f times −2," and "(f∘g) means just multiply f and g."
✅ Cure: f(x) is a machine, not a product — f(−2) is the output at −2. And composition is nesting, not multiplying: (f∘g)(x) = f(g(x)), inside first. "Functions are machines; composition feeds one machine's output into the next." (And for a domain, the question is always "what would make this illegal?" — a zero denominator or a negative under a square root.)

---

Segment 4 — Objective 4 Review: Linear Functions, Graphs & Systems (29 min) · Session 1 closes (~75)

Re-teach lines in plain language. A line is the graph of a constant rate of change. Three moves:
- Slope between (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁) — "rise over run." Sign tells direction; size tells steepness.
- Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis). Given a slope and the intercept, you can write the line immediately.
- Parallel/perpendicular: parallel lines share a slope; perpendicular slopes are negative reciprocals (m and −1/m).

Re-teach systems in plain language. A system asks for the point that satisfies both equations at once — the intersection. Substitution (solve one equation for a variable, plug into the other) and elimination (add/subtract to cancel a variable) both work; pick whichever is cleaner.

One quick worked example (compute it live):

Find the slope through (2, −1) and (5, 8); write the line with slope 2 through (0, −3); solve the system 2x + y = 5, x − y = 1.
- Slope: (8 − (−1))/(5 − 2) = 9/3 = 3. (Watch the double negative in the numerator.)
- Line: slope 2, y-intercept −3 → y = 2x − 3.
- System: add the two equations (the y's cancel): 3x = 6 → x = 2; then 2 + y = 5 → y = 1 → (2, 1). Check in both. ✓

Highest-cost misconception + cure:
- ❌ "Slope is run over rise," and "the solution to a system is just one number."
✅ Cure: slope is rise over run — change in y over change in x; flipping it is the most common slope error, along with mishandling the minus signs in the differences. And a 2×2 system's solution is an ordered pair (x, y) — the point where the lines meet — not a single value. "Slope = rise over run; a system's answer is a point."

Quick interaction (think-pair-share, ~6 min): put four prompts on a slide; students call the move (slope / write-a-line / substitution / elimination) solo (30 s), neighbor (1 min), vote by fingers. (e.g., "two points given" → slope; "one equation already says y = …" → substitution; "the y-terms are +y and −y" → elimination.)

---

Segment 5 — Objective 5 Review: Polynomials & Factoring (22 min) · Session 2 opens

Hook back in: "Session 1 we simplified, solved, built functions, and graphed lines. Now we take expressions apart — factoring is multiplication run backwards, and it's the key that unlocks quadratics next."

Re-teach in plain language. Factoring rewrites a sum as a product. The order to try:
- GCF first, always: pull the greatest common factor out of every term (e.g., 6x³ − 9x² = 3x²(2x − 3)).
- Trinomial x² + bx + c: find two numbers that multiply to c and add to b; they become the constants in (x + ?)(x + ?).
- Difference of squares: a² − b² = (a − b)(a + b) — only for a subtraction of two perfect squares (no middle term).
- Perfect-square trinomial: a² + 2ab + b² = (a + b)² (and a² − 2ab + b² = (a − b)²).

One quick worked example (compute it live):

Factor x² − 5x − 14.
- Two numbers that multiply to −14 and add to −5: that's −7 and +2 (−7 · 2 = −14, −7 + 2 = −5).
- So x² − 5x − 14 = (x − 7)(x + 2). Check by expanding (FOIL): x² + 2x − 7x − 14 = x² − 5x − 14. ✓
- Contrast the difference of squares: x² − 16 = (x − 4)(x + 4), and the perfect square (x + 3)² = x² + 6x + 9.

Highest-cost misconception + cure:
- ❌ "Skip the GCF," and "every difference factors as a difference of squares."
✅ Cure: always pull the GCF first — factoring it later (or never) is the most common lost-points move. And difference of squares needs two perfect squares with a minus (x² − 16 ✓); a sum of squares (x² + 16) does not factor over the reals, and x² − 5x − 14 is a trinomial, not a difference of squares. "GCF first; difference of squares only when both pieces are perfect squares and it's a subtraction."

---

Segment 6 — Objective 6 Review: Quadratics — Equations, Functions & Graphs (20 min)

Re-teach in plain language. A quadratic is degree two; its graph is a parabola. Three moves:
- Solve by factoring + the zero-product property: if a product equals 0, at least one factor is 0. Factor, set each factor to 0, solve. (x² − 4x − 5 = 0 → (x − 5)(x + 1) = 0 → x = 5 or x = −1.)
- Read the vertex from vertex form: f(x) = a(x − h)² + k has vertex (h, k) — note the sign flip on h (the form has minus h). The vertex is the parabola's max or min.
- Count real solutions with the discriminant: b² − 4ac. Positive → two real solutions; zero → one; negative → no real solutions (the parabola doesn't cross the x-axis).

One quick worked example (compute it live):

Solve x² − 4x − 5 = 0; read the vertex of f(x) = (x − 1)² − 4; find the discriminant of x² + 2x + 5.
- Solve: factor to (x − 5)(x + 1) = 0 → x = 5 or x = −1. Check both.
- Vertex: f(x) = (x − 1)² − 4 is vertex form with h = 1, k = −4 → vertex (1, −4) (the x-coordinate is +1, not −1).
- Discriminant: b² − 4ac = 2² − 4(1)(5) = 4 − 20 = −16 → negative → no real solutions.

Highest-cost misconception + cure:
- ❌ "If (x − 5)(x + 1) = 0 then x = 5 and the other factor doesn't matter," and "the vertex of (x − 1)² − 4 is (−1, −4)."
✅ Cure: the zero-product property gives every solution — set each factor to 0 (x = 5 and x = −1). And in a(x − h)² + k the vertex x is +h — the form already has the minus built in, so (x − 1)² has vertex x = +1. "Every factor gives a root; the vertex flips the sign you see inside the parentheses." (And a negative discriminant means no real roots — don't force a real answer.)

---

Segment 7 — Objective 7 Review: Rational & Radical Expressions and Equations (24 min)

Re-teach the rational/radical pieces in plain language. Objective 7 extends the toolkit to fractions-with-variables and roots:
- Simplify a rational expression by factoring top and bottom and cancelling common factors — and note the restriction (the value that made the original denominator zero is still excluded). (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3, x ≠ 3.
- Rational exponents: a^(m/n) = (ⁿ√a)ᵐ — the denominator is the root, the numerator is the power. (27^(2/3) = (³√27)² = 3² = 9.)
- Solve a radical equation by isolating the radical and raising both sides to the matching power, then checking for extraneous solutions (squaring can introduce false answers). (√(x + 1) = 4 → x + 1 = 16 → x = 15; check √16 = 4 ✓.)

One quick worked example (compute it live):

Simplify (x² − 9)/(x − 3); evaluate 27^(2/3); solve √(x + 1) = 4.
- Simplify: factor the numerator as a difference of squares → (x − 3)(x + 3)/(x − 3) = x + 3, with the restriction x ≠ 3 carried along.
- Rational exponent: 27^(2/3) = (³√27)² = 3² = 9 (cube root first, then square).
- Radical equation: square both sides → x + 1 = 16 → x = 15; check: √(15 + 1) = √16 = 4 ✓ (not extraneous).

Highest-cost misconception + cure:
- ❌ "Cancel the x's: (x² − 9)/(x − 3) = x − 3," "27^(2/3) = 18," and "skip the check on a radical equation."
✅ Cure: you can only cancel common factors, not loose terms — factor first, then cancel the matching (x − 3), giving x + 3. For a rational exponent, the bottom is the root: 27^(2/3) is the cube root (3) squared = 9, not (2/3)·27. And always check radical solutions — squaring both sides can create extraneous roots. "Factor before you cancel; bottom-is-the-root; and every radical answer gets checked."

---

Segment 8 — Objectives 8 Review + the Final Frame: Exponentials, Logs & How to Prepare (24 min) · Session 2 closes (~75)

Re-teach Obj 8 in plain language (one last big idea). An exponential function (bˣ, or eˣ for continuous growth) models things that grow or shrink by a constant factor. A logarithm is its inverse — it answers "what exponent?":

logb(x) = y means 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 is 16?" → 4 (since 2⁴ = 16). log₂8 = 3, log₅1 = 0 (anything to the 0 is 1), ln e = 1.
- The inverse relationship is the whole key: every log fact is an exponential fact in disguise, and that's what lets you solve "solve for the exponent" problems (Week 15's compound-interest and half-life work).

One quick worked example (compute it live):

Evaluate log₂16, and decide which of these are true: log₂8 = 3 · log₅1 = 0 · 2³ = 6 · ln e = 1 · log₁₀100 = 10.
- log₂16 = 4 (2⁴ = 16).
- True: log₂8 = 3 (2³ = 8) ✓; log₅1 = 0 (5⁰ = 1) ✓; ln e = 1 (e¹ = e) ✓.
- False: 2³ = 8, not 6 ✗; log₁₀100 = 2 (10² = 100), not 10 ✗.

Highest-cost misconception + cure:
- ❌ "log₂8 = 4," "log₅1 = 5," and "ln e = 0."
✅ Cure: a log is the exponent — log₂8 = 3 because 2³ = 8; logb(1) = 0 for any base (b⁰ = 1); and ln e = 1 because e¹ = e. "A log is the power; log of 1 is 0; ln e is 1." Turn every log into its exponential form and the value is obvious.

Audit-the-AI review moment (the course's recurring habit, one last time before the exam):

Paste to an approved chatbot: "Is −3² equal to 9? Also, does √(x + 1) = 4 give x = 15, and do I need to check it? And is log₁₀100 = 10?"
Check it against what we taught. Chatbots often (1) call −3² = 9 (it's −9 — the exponent binds tighter than the sign), (2) skip the extraneous-solution check on a radical equation, and (3) misread log₁₀100 as 10 (it's 2). The tool drafts; you judge. Catch all three and you're ready. (Reminder: AI is for prep only — the Final itself is closed to AI.)

What's on the Final (state it plainly — put it on the closing slide):
- Coverage: cumulative over the whole course — Weeks 1–15, Objectives 1–8. Simplifying; solving equations & inequalities; functions; lines & systems; factoring; quadratics; rational & radical expressions/equations; and exponential & logarithmic functions. Weighted toward the function/quadratic/rational-radical/exponential-log material since the midterm already covered Objectives 1–6 as taught in Weeks 1–7 — but the early objectives are tools the later ones use, so they're fair game.
- Weight & logistics: the Final is 30% of the course grade. The window opens Mon Dec 14 and the exam is due Fri Dec 18, 11:59 p.m. (end of finals). No AI on the Final. (There is no Quiz 16, no Discussion 16, and no Assignment 16 — the Final replaces all of them.)
- Format: 20 auto-graded items, 100 points (5 each) — a mix of simplify / solve / evaluate / interpret in the spirit of the worked examples above, spread across all eight objectives.

The preparation plan (point at each artifact by name):
1. Study Guide — work it first; it's the checklist of every move across the eight objectives.
2. Exam-Prep Tutorial — run it with an approved chatbot (Gemini / Claude / ChatGPT) and submit the share link; it drills your weak spots adaptively.
3. Practice Final — sit it timed, like the real thing, then review every miss against the Study Guide.

Callback + send-off:
- Callback: "Every item on the Final is a move you already made this term — Week 1 you learned to respect the rules that never change, and that instinct runs through all eight objectives: simplify without changing the value, solve cleanly, build and graph the objects, and model how the world grows and shrinks."
- Send-off: "You don't need to cram everything — you need the eight honest moves and the mistake that sinks each one. Work the Study Guide, run the Exam-Prep Tutorial, take the Practice Final, then sit the Final. You've built every one of these skills. Go show them."

Hand-off (the week's work): review the Study Guide, run the Exam-Prep Tutorial (submit the share link), take the Practice Final, and sit the comprehensive Final (window opens Mon Dec 14; due Fri Dec 18). No quiz, discussion, or assignment this week — the Final is the whole grade for the module.

---

Instructor FAQ — Common Stumbles (Final-Review Week)

Student says / does	Quick cure
Writes −4² = 16.	The exponent binds tighter than the sign: −4² = −16 (only the 4 is squared). You get +16 only inside parentheses, (−4)².
Multiplies the exponents in x²·x³.	Same base times same base → add the exponents → x⁵. You multiply exponents only for a power of a power, (x²)³ = x⁶.
Forgets to flip the inequality when dividing by a negative.	Multiplying/dividing an inequality by a negative reverses the sign. It's the single most common inequality error.
Reads f(−2) as "f times −2."	f(x) is a machine, not a product. f(−2) means "substitute −2 for x and simplify."
Gives a domain without excluding anything.	Ask what would be illegal? — a zero denominator (set it ≠ 0) or a negative under a square root (set the inside ≥ 0).
Computes slope as run over rise.	Slope is rise over run = change in y ÷ change in x. And mind the minus signs in the differences.
Stops a system at a single number.	A 2×2 system's solution is an ordered pair (x, y) — the intersection point. Solve for both and check in both equations.
Skips the GCF when factoring.	Always pull the GCF first. Factoring it later (or not at all) loses points and hides the structure.
Factors x² + 16 as a difference of squares.	A sum of squares doesn't factor over the reals. Difference of squares needs two perfect squares with a minus (x² − 16 ✓).
From (x − 5)(x + 1) = 0, gives only x = 5.	The zero-product property gives every root: set each factor to 0 → x = 5 and x = −1.
Says the vertex of (x − 1)² − 4 is (−1, −4).	In a(x − h)² + k the vertex x is +h = +1. The minus is already in the form.
Cancels terms: (x² − 9)/(x − 3) = x − 3.	Factor first, then cancel common factors: (x − 3)(x + 3)/(x − 3) = x + 3, x ≠ 3. You can't cancel loose terms.
Evaluates 27^(2/3) = 18.	A rational exponent: bottom is the root, top is the power → (³√27)² = 3² = 9.
Skips the check on a radical equation.	Squaring can introduce extraneous solutions. Always substitute your answer back into the original radical equation.
Says log₂8 = 4 or ln e = 0.	A log is the exponent: log₂8 = 3 (2³ = 8); ln e = 1 (e¹ = e); and logb 1 = 0 for any base.
Panics that the Final is "literally everything."	It's the eight honest moves, not a thousand problems. The function/quadratic/rational-radical/exp-log material leans heaviest; the early objectives are the tools. Work the Study Guide → Exam-Prep Tutorial → Practice Final, in that order.

---

Scope flag

This outline is pure review of Objectives 1–8 — no new material. The framing extras (the "language → objects → toolkit" three-act map and the recurring −4² / GCF-first / extraneous-check callbacks) are retained context carried over from the term because they make the cures stick; cut them for a leaner 60-minute review. The Final and its bundle (Study Guide, Exam-Prep Tutorial, Practice Final) are built separately and only referenced here by name. No quiz, discussion, or assignment is built for Week 16 — by the course spine, discussions run every week except W16, and exam weeks replace the quiz and assignment with the exam; the comprehensive Final is the module's only graded item.