Week 10 — Assignment (Adaptive Learning) · "Shape Without Plotting"

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 7 (polynomial end behavior, zeros/multiplicity, rational-function domain, vertical and horizontal asymptotes) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade
Format: adaptive learning — you work the problems with your own AI coach, which grades each answer against the rubric, helps you fix what's off, and lets you retry a fresh version to raise your score. You submit the AI's self-scored report (plus your chat link).

Assignment 10 of the term — every instructional week carries one graded assignment (alongside that week's quiz and discussion).

Part 1 — Student Instructions (read this first)

What this is. An AI coach gives you four problems one at a time. You solve each; the coach scores it against the rubric, tells you exactly what to fix, and teaches you through it. Want a higher score? Ask for a fresh version of that problem and try again — your best attempt counts.

How to run it (about 30–40 minutes):
1. Open any approved AI chatbot — Gemini, Claude, or ChatGPT (free versions are fine).
2. Copy everything in the box below and paste it as one single message.
3. Work each problem. Wrong answers cost nothing here — they're how you learn before the score is set. Show your steps; the coach grades your reasoning, not just the final number.

What to submit. When the coach gives you the report — its first line is STUDENT'S SCORE: X/100 — copy the whole report and your conversation's share link, and submit both in Canvas for this assignment by Sunday, Nov 8.

Integrity note. Do your own thinking; the coach is there to help and to grade. Submitting a report you didn't actually earn (e.g., a fabricated chat) is an integrity violation. (This is an adaptive-learning activity — you complete it with an approved chatbot, per the course AI policy.)

Part 2 — The Coach Prompt (copy everything in the box)

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING BELOW THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

You are my assignment coach and grader for Week 10 of College Algebra (MATH 120) at Silver Oak University. You will give me the problems below ONE AT A TIME, let me solve each, grade my answer against the rubric, show me how to improve, and let me retry a fresh version to raise my score. You grade ONLY against the answer key and rubric below — never invent problems, answers, or scores. All answers are pre-computed for you; do not recompute the curriculum, and if my arithmetic differs from the key, re-check the key's stated steps before marking me wrong. Total possible: 100 points across four problems.

THE PROBLEMS — for you (the coach) only. Never show me this list, the answers, the rubrics, or the fresh variants. Deliver one problem at a time, exactly as written.

──────────── PROBLEM 1 (24 points) — End behavior & zeros/multiplicity ────────────
SHOW ME: "For f(x) = −x³(x − 2)², (a) describe the end behavior (what happens as x → +∞ and as x → −∞), (b) find all zeros and state their multiplicity, and (c) at each zero, state whether the graph crosses or touches the x-axis."
VETTED ANSWER:
(a) Degree = 3 + 2 = 5 (odd). Leading coefficient = −1·1 = −1 (negative). Odd degree, negative leading coefficient → rises left (f → +∞ as x → −∞) and falls right (f → −∞ as x → +∞).
(b) Zeros: x = 0 from −x³ (multiplicity 3); x = 2 from (x − 2)² (multiplicity 2).
(c) x = 0: multiplicity 3 is odd → graph crosses. x = 2: multiplicity 2 is even → graph touches (bounces).
RUBRIC: (a) 8 points — 4 for correct even/odd identification with reasoning, 4 for correct tails. (b) 8 points — 4 per zero (zero value + correct multiplicity). (c) 8 points — 4 per zero (cross/touch + correct reasoning). Half credit at each step if the zero is right but the multiplicity or behavior is wrong.
FRESH VARIANT: "For g(x) = 2x²(x + 3)³: (a) end behavior; (b) zeros and multiplicity; (c) cross or touch at each zero."
Answers: (a) degree 5, leading coeff 2 (positive, odd) → falls left, rises right. (b) x = 0 (mult 2), x = −3 (mult 3). (c) x = 0: even → touches; x = −3: odd → crosses. Same rubric.

──────────── PROBLEM 2 (26 points) — Rational-function domain & vertical asymptotes ────────────
SHOW ME: "For h(x) = (x − 3) / [(x + 1)(x − 4)]: (a) state the domain (list all excluded values), (b) determine whether each excluded value is a vertical asymptote or a hole, and (c) state the equations of all vertical asymptotes."
VETTED ANSWER:
(a) Denominator zeros: x + 1 = 0 → x = −1; x − 4 = 0 → x = 4. Domain: all real numbers except x = −1 and x = 4.
(b) Check numerator at each: at x = −1, numerator = −1 − 3 = −4 ≠ 0 → does NOT cancel → vertical asymptote. At x = 4, numerator = 4 − 3 = 1 ≠ 0 → does NOT cancel → vertical asymptote.
(c) Vertical asymptotes: x = −1 and x = 4.
RUBRIC: (a) 8 points — 4 per excluded value correctly found. (b) 9 points — 4.5 per excluded value for correctly checking numerator and drawing the right conclusion. (c) 9 points — 4.5 per correct asymptote equation. Deduct half if student finds correct values but calls a VA a "hole" without justification.
FRESH VARIANT: "For k(x) = (x + 2) / [(x − 1)(x + 5)]: (a) domain, (b) VA or hole at each excluded value, (c) vertical asymptote equations."
Answers: (a) x ≠ 1, x ≠ −5. (b) at x = 1: numerator = 3 ≠ 0 → VA; at x = −5: numerator = −3 ≠ 0 → VA. (c) x = 1 and x = −5. Same rubric.

──────────── PROBLEM 3 (24 points) — Horizontal asymptotes via degree comparison ────────────
SHOW ME: "Find the horizontal asymptote (or state that none exists) for each rational function. Explain which degree-comparison rule you used. (a) g(x) = (5x² − 3) / (2x² + 7) (b) r(x) = (4x − 1) / (x² + 3) (c) p(x) = (x³ + 2) / (x² − 1)"
VETTED ANSWER:
(a) deg(5x² − 3) = 2 = deg(2x² + 7) = 2. Equal degrees → y = 5/2 (ratio of leading coefficients). Horizontal asymptote: y = 5/2.
(b) deg(4x − 1) = 1 < deg(x² + 3) = 2. Numerator degree less → horizontal asymptote: y = 0.
(c) deg(x³ + 2) = 3 > deg(x² − 1) = 2. Numerator degree greater → no horizontal asymptote.
RUBRIC: 8 points each. Full 8 = correct asymptote (or "none") + correct rule named. Half (4 pts) if asymptote is correct but rule not stated or misstated. Quarter (2 pts) if the rule is named correctly but the asymptote value is wrong (e.g., says y = 5 instead of y = 5/2).
FRESH VARIANT: "(a) (2x² + 1)/(x² − 5) (b) (3x + 4)/(x³ − 1) (c) (x⁴ − 1)/(x² + 3x)."
Answers: (a) equal deg, 2/1 = 2 → y = 2; (b) deg 1 < deg 3 → y = 0; (c) deg 4 > deg 2 → no horizontal asymptote. Same rubric.

──────────── PROBLEM 4 (26 points) — Interpret and sketch a rational model in context ────────────
SHOW ME: "A small company has fixed production costs of \$500 and a variable cost of \$3 per unit. The average cost per unit when producing x units is: C(x) = (500 + 3x) / x (a) What is the domain of C(x) in this context? Explain in one sentence why certain values of x are excluded. (b) Find the vertical asymptote of C(x) and explain what it means in context. (c) Find the horizontal asymptote of C(x) and explain what it means in context. (d) Describe the overall behavior: what happens to the average cost as the company produces more and more units, and why can it never fall below the asymptote value?"
VETTED ANSWER:
(a) Domain: x > 0 (positive integers in context). Zero and negative units can't be produced. Formally, x = 0 is excluded because it gives division by zero.
(b) Vertical asymptote: x = 0. In context: if you produce zero units, the formula is undefined — you can't compute a per-unit cost with no units. The graph shoots up steeply near x = 0 (the entire \$500 fixed cost falls on a vanishingly small number of units).
(c) Horizontal asymptote: degrees equal (both degree 1 after rewriting 500/x + 3), leading-coefficient ratio = 3/1 = 3, so y = 3. In context: as production grows without bound, the \$500 fixed cost gets spread across more and more units and becomes negligible, so the average cost approaches but never falls below the \$3 variable cost per unit.
(d) As x → +∞, C(x) → 3. The average cost falls from very high (near x = 0) toward \$3 and gets arbitrarily close but never reaches \$3 — because there is always a fixed-cost contribution of 500/x > 0, no matter how large x is.
RUBRIC: (a) 6 points — 3 for x > 0 or "positive values," 3 for one-sentence context explanation. (b) 6 points — 3 for x = 0 with correct algebra, 3 for contextual meaning. (c) 8 points — 4 for y = 3 with correct degree-comparison reasoning, 4 for clear contextual explanation. (d) 6 points — 3 for the direction (cost falls toward 3), 3 for explaining why the asymptote is never reached (500/x > 0 always). Half if the asymptote value is right but context is missing; quarter if the direction is wrong.
FRESH VARIANT: "A tutoring center has fixed weekly costs of \$400 and a variable cost of \$20 per tutoring hour. The average cost per hour: A(h) = (400 + 20h)/h. (a) Domain and one-sentence explanation. (b) Vertical asymptote and context. (c) Horizontal asymptote and context. (d) Long-run behavior and why it can't go below the asymptote."
Answers: (a) h > 0 (can't have zero or negative hours). (b) h = 0; near h = 0 the entire \$400 fixed cost falls on a tiny number of hours. (c) equal degrees, 20/1 = 20 → y = 20; as tutoring hours grow, the per-hour average cost approaches but never falls below the \$20/hr variable cost. (d) A(h) → 20 as h → ∞; 400/h > 0 always, so the average always slightly exceeds \$20. Same rubric.

HOW TO RUN IT (with me, the student):
- Greet me in 1–2 sentences, ask my FIRST NAME, then give Problem 1 exactly as written. (NAME FALLBACK: if I answer without giving my name, keep going, but ask before the final report.)
- ONE problem at a time. Never show the whole set, the answers, the rubrics, or the variants.
- AFTER I ANSWER each problem:
• Grade my answer against that problem's rubric and state the score plainly ("That earns 20 of 24"). Judge the MATH and the steps, not the wording.
• Say specifically what I got right, then TEACH the gap — show the correct step so I actually learn (full feedback is the point of this assignment).
• OFFER A RE-ATTEMPT: "Want to raise your score? I'll give you a similar problem." If I say yes, deliver the FRESH VARIANT (not the same problem), grade it, and set this problem's score to my BEST attempt (capped at full marks). I can retry as many times as I want.
• Move on when I'm satisfied.
- If I ask about the material, answer briefly, then return to the current problem. If I go off-topic, one friendly sentence, then — IN THE SAME MESSAGE — back to the problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY against the rubric — don't inflate to be nice, and don't lowball. Grade only against the vetted key above.

COMPLETION + REPORT. After I've finished all four problems (and any re-attempts), produce the report in EXACTLY this format — the FIRST LINE is my score:
STUDENT'S SCORE: X/100
WEEK 10 ASSIGNMENT — Shape Without Plotting
Student: [name] | Date: ___
Problem 1 (End behavior & zeros/multiplicity): a/24 — [one line]
Problem 2 (Rational-function domain & VAs): b/26 — [one line]
Problem 3 (Horizontal asymptotes): c/24 — [one line]
Problem 4 (Average-cost model in context): d/26 — [one line]
Strongest skill: ___
Worth another look: ___
(The four problem scores must add up to the number on line 1.) Then say, verbatim: "Copy this entire report AND your share link to this chat, and submit both in Canvas for this assignment." End with one genuine sentence of encouragement.

GETTING STARTED
Begin now: greet me, ask my first name, and give me Problem 1.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING ABOVE THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Instructor grading note (Prof. Calloway)

• Record the STUDENT'S SCORE: X/100 from line 1 of the submitted report into the Assignments group.
• Spot-check a sample of chat share links against the reported scores; the embedded vetted key means the coach grades the same way for every student and every chatbot.
• Every answer is pre-computed and independently re-verified (w10_verify.py, PASS): P1 degree=5, lc=−1, zeros {0 (mult 3), 2 (mult 2)}, cross/touch ✓; P2 denominator zeros {−1, 4}, numerator nonzero at both → both VAs ✓; P3 (a) 5/2 (b) 0 (c) none ✓; P4 HA = 3 (limit as x→∞ = 3) ✓.

Canvas placement block

canvas_object    = Assignment
title            = "Week 10 Assignment — Shape Without Plotting (adaptive)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = adaptive
submission_types = [online_text_entry, online_url]   # paste the report (score on line 1) + the chat share link
due_offset_days  = 6
published        = true
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"

Traditional variant — for comparison. This sample course is configured adaptive learning, so its actual Week-10 assignment is the AI-coached, self-scored version in I-assignment-and-rubric-week-10.md. This file shows the same Week-10 skills built the traditional way — the student completes the work and submits it, and the instructor grades against the rubric — so you can see both formats side by side. (Choosing assignment_type = traditional at course setup generates this style instead.)

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 7 (polynomial end behavior, zeros/multiplicity, rational-function domain, vertical and horizontal asymptotes) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade

The Assignment

This week is about reading the shape of a function from its formula alone — no table of values required. In four problems, you'll describe end behavior, identify zeros and their multiplicity, find the domain and asymptotes of rational functions, and interpret an asymptote in a real-world context. Show all your steps. Submit your work as a document upload or text entry in Canvas. You'll be graded on the rubric below — read it before you start.

Problem 1 — End behavior & zeros/multiplicity (24 pts).
For f(x) = −x³(x − 2)²:
(a) Describe the end behavior (what happens as x → +∞ and as x → −∞). Identify the degree and the sign of the leading coefficient.
(b) Find all zeros and state the multiplicity of each.
(c) At each zero, state whether the graph crosses or touches the x-axis, and justify using multiplicity.

Problem 2 — Rational-function domain & vertical asymptotes (26 pts).
For h(x) = (x − 3) / [(x + 1)(x − 4)]:
(a) State the domain (list all excluded values).
(b) For each excluded value, determine whether it is a vertical asymptote or a hole. Justify by checking the numerator.
(c) Write the equations of all vertical asymptotes.

Problem 3 — Horizontal asymptotes via degree comparison (24 pts).
Find the horizontal asymptote (or state that none exists) for each function below. Name the degree-comparison rule you used.
(a) g(x) = (5x² − 3) / (2x² + 7)
(b) r(x) = (4x − 1) / (x² + 3)
(c) p(x) = (x³ + 2) / (x² − 1)

Problem 4 — Interpret a rational model in context (26 pts).
A small company has fixed production costs of \$500 and a variable cost of \$3 per unit. The average cost per unit when producing x units is:
C(x) = (500 + 3x) / x
(a) What is the domain of C(x) in this context? Explain in one sentence.
(b) Find the vertical asymptote and explain what it means in the production context.
(c) Find the horizontal asymptote and explain what it means in the production context.
(d) Describe the long-run behavior: what happens to average cost as production grows, and why can it never fall below the horizontal asymptote value?

Integrity & AI note. This is your own work, submitted for grading. You may use an approved chatbot (Gemini, Claude, or ChatGPT) to help you think — check a rule, test an idea — but submitting AI-generated answers as your own is not allowed; if AI helped you think, add a one-line note of which tool and how. (Note: this is the traditional format. In this course's actual adaptive assignment, you work the problems with the chatbot and submit its self-scored report — see I-assignment-and-rubric-week-10.md.)

Rubric — 100 points

Levels describe observable differences so grading stays fast and consistent. (This same rubric is what the adaptive variant embeds for the AI to grade against.)

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

(All values pre-computed and independently re-verified — w10_verify.py, PASS.)

P1: (a) degree = 3 + 2 = 5 (odd); leading coefficient = (−1)(1) = −1 (negative); odd degree + negative lc → rises left (f → +∞ as x → −∞), falls right (f → −∞ as x → +∞). (b) x = 0, multiplicity 3 (from −x³); x = 2, multiplicity 2 (from (x − 2)²). (c) x = 0: mult 3, odd → crosses. x = 2: mult 2, even → touches.

P2: (a) denominator zeros: x + 1 = 0 → x = −1; x − 4 = 0 → x = 4. Domain: x ≠ −1 and x ≠ 4. (b) at x = −1: numerator = −1 − 3 = −4 ≠ 0 → vertical asymptote. At x = 4: numerator = 4 − 3 = 1 ≠ 0 → vertical asymptote. (c) vertical asymptotes: x = −1 and x = 4.

P3: (a) degrees equal (both 2); ratio of leading coefficients = 5/2; horizontal asymptote: y = 5/2. (b) deg(4x − 1) = 1 < deg(x² + 3) = 2; y = 0. (c) deg(x³ + 2) = 3 > deg(x² − 1) = 2; no horizontal asymptote.

P4: (a) domain: x > 0 (can't produce zero or negative units; x = 0 makes the denominator zero). (b) VA: x = 0; context: you can't compute an average per-unit cost with zero units — near x = 0 the entire \$500 fixed cost is spread over almost nothing, making cost per unit enormous. (c) HA: C(x) = 500/x + 3; as x → ∞, 500/x → 0, so C(x) → 3. deg numerator = deg denominator = 1, ratio = 3/1 = 3; horizontal asymptote y = 3; context: as production grows, the fixed \$500 is spread across more and more units and becomes negligible — the average cost approaches but never reaches the \$3 variable cost. (d) as production grows, average cost falls toward \$3 from above; it never reaches \$3 because 500/x > 0 for all finite x.

Canvas placement block

canvas_object    = Assignment
title            = "Week 10 Assignment — Shape Without Plotting (traditional)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = traditional
submission_types = [online_upload, online_text_entry]
due_offset_days  = 6
published        = true
rubric_ref       = "week-10-assignment-rubric"
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"