Week 10 — Read the Graph Before You Plot: A Short Report on Polynomial and Rational Functions

Assignment 9 of the term — read four graph features straight from their formulas, explain what each one shows, and tell someone new to algebra what a horizontal asymptote means for a real cost.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 3 you learned that a rational function's domain leaves out every zero of its denominator, and in Week 6 you factored polynomials; this week you read a graph's ends, zeros, and asymptotes straight from those factored formulas. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

You will use four functions from this week: the polynomial x²(x − 3)(x + 2)³, the polynomial f(x) = (x − 1)²(x + 4), the rational function R(x) = (x + 1) ⁄ (x² − 4), and a print shop's average cost per tote bag. The shop pays 200 dollars once to set up its press and 5 dollars of material per bag, so making x bags costs 5x + 200 dollars and the average cost per bag is u(x) = (5x + 200) ⁄ x.

Work these four problems, showing your arithmetic:

1. The polynomial x²(x − 3)(x + 2)³ is written as a product of factors. What is its degree?
2. For f(x) = (x − 1)²(x + 4), what is f(−2)?
3. For R(x) = (x + 1) ⁄ (x² − 4), what is R(3)?
4. A print shop pays 200 dollars to set up plus 5 dollars per tote bag, so its average cost per bag is u(x) = (5x + 200) ⁄ x. What is u(50), in dollars?

Beside each answer, add one brief interpretation: what that value shows about the graph and one use or limit of it. For problem 1, use the degree and the leading coefficient to describe both ends, and say whether the graph crosses or touches at each zero. For problem 2, test a value on the other side of −4 and explain why the graph crosses at −4 but only touches at 1. For problem 3, name the vertical and horizontal asymptotes and say what the sign of R(3) shows about the graph to the right of x = 2. For problem 4, name the horizontal asymptote and explain what it means for the shop's average cost.

Explain it to a non-expert

The print shop's owner has never studied algebra. They ask: "If I make enough bags, will my average cost finally get down to 5 dollars a bag?" Write a 3–5 sentence message for the owner: what happens to the average cost as they make more bags, why it gets close to 5 dollars but never reaches it, and one honest limit of the model, such as what it assumes about the material cost. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine a community pottery studio that charges a one-time 120-dollar membership plus 8 dollars of clay for each piece you make. Model answer: "Every piece always needs its 8 dollars of clay, and the 120-dollar membership gets split over more pieces the more you make. So your average cost per piece keeps dropping toward 8 dollars, but it never quite gets there, because a small share of the membership is always included. The formula assumes the clay costs the same for every piece, so one very large piece would break the pattern."

Use that pattern for the print shop, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for the shop's owner before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about polynomial and rational functions: the leading term and end behavior, zeros and multiplicity, vertical asymptotes and holes, and horizontal asymptotes and what they mean in a setting. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the polynomial x^2(x - 3)(x + 2)^3, the polynomial f(x) = (x - 1)^2(x + 4), the rational function R(x) = (x + 1) / (x^2 - 4), and a print shop that pays 200 dollars once to set up plus 5 dollars per tote bag, so its average cost per bag is u(x) = (5x + 200) / x.
PROBLEM P1: The polynomial x²(x − 3)(x + 2)³ is written as a product of factors. What is its degree?
PROBLEM P2: For f(x) = (x − 1)²(x + 4), what is f(−2)?
PROBLEM P3: For R(x) = (x + 1) ⁄ (x² − 4), what is R(3)?
PROBLEM P4: A print shop pays 200 dollars to set up plus 5 dollars per tote bag, so its average cost per bag is u(x) = (5x + 200) ⁄ x. What is u(50), in dollars?
After the four answers, I must briefly interpret each one: what that value shows about the graph and one use or limit of it; for P1, the end behavior from the degree and the leading coefficient and whether the graph crosses or touches at each zero; for P2, a test value on the other side of -4 and why the graph crosses at -4 but only touches at 1; for P3, the vertical and horizontal asymptotes and what the sign of R(3) shows; for P4, the horizontal asymptote and what it means for the shop's average cost. I must then write a 3-5 sentence message for the shop's owner, who is new to algebra, explaining what happens to the average cost as they make more bags, why it gets close to 5 dollars but never reaches it, and one honest limit of the model. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct function computations (40 points), reasoning about graph features and the setting (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct degree, value f(-2), value R(3), or average cost u(50) = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what that value shows about the graph plus a use or limit = 2; exactly one of the two = 1; neither = 0. For P1, a degree read from the largest exponent alone does not count as the account; for P2, calling the zero at 1 a crossing does not count as the account. For P4, BOTH an accurate explanation that the horizontal asymptote is a level the average cost approaches but never reaches AND a final message that explains that floor with an honest limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message for the shop's owner AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised function report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised function report with all four answers, their brief interpretations, and your message for the shop's owner — written, or recorded as audio or video with a transcript, or as a labelled sketch with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 10 skills — reading a polynomial's degree, end behavior, and zeros with their multiplicities, finding a rational function's vertical and horizontal asymptotes, and explaining what a horizontal asymptote means in a setting — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message for the shop's owner. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised function report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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