Week 10 — Read the Graph Before You Plot: A Short Report on Polynomial and Rational Functions — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned to read a polynomial's end behavior from its leading term, to find its zeros and decide from each multiplicity whether the graph crosses or touches, to sort a rational function's excluded inputs into vertical asymptotes and holes, and to find a horizontal asymptote by comparing degrees. 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 straight from those factored formulas. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied algebra what a horizontal asymptote means for a real cost. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single function report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows about the graph and one use or limit of it;
• your message for the shop's owner, 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; and
• a one-line restatement of the functions you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your answers. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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