Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Module: Week 10 of 16 · Fall 2026 · in-person, two 75-minute sessions
Objective covered: Objective 7 — Analyze polynomial and rational functions using end behavior, zeros and multiplicity, domain, and vertical and horizontal asymptotes.

This file holds two pieces: (A) the Module 10 Overview page ("Start Here") and (B) the Week 10 Announcement that drips out when the module opens. Dates below assume a Tuesday/Thursday session pattern with Week 10 meeting Tue Nov 3 and Thu Nov 5, and end-of-week work due Sunday Nov 8, 11:59 p.m.

(A) Module 10 Overview — Start Here

Welcome to Week 10: Polynomial & Rational Functions

This is your home base for the week. Read it first, then work the checklist below from top to bottom. Everything you need is linked inside the module.

Last week you mastered the parabola — one specific polynomial function. This week we zoom out and ask: what can we say about the shape and behavior of any polynomial, and then of the messier cousins called rational functions? The answers come from two powerful ideas: where the leading term dominates at the edges (end behavior) and where the function crashes or grazes the x-axis (zeros and multiplicity). Add in asymptotes for rational functions and you have a complete graphing toolkit — no plotting required.

The week's big question

"Without plotting a single point, what can the leading term, the zeros, and the asymptotes tell you about the entire shape of a function's graph?"

By the end of this week you will read a polynomial and immediately know which way its ends point and where its graph crosses or bounces off the x-axis — and you will find the domain, vertical asymptotes, and horizontal asymptote of a rational function from the algebra alone.

By the end of this week, you can…

Use this as a checklist. If you can do all four, you're ready for the quiz.

• [ ] Describe end behavior of any polynomial from its leading term: odd/even degree tells you symmetry of the tails; the sign of the leading coefficient tells you up or down.
• [ ] Identify zeros and multiplicity of a polynomial in factored form; state whether the graph crosses (odd multiplicity) or touches and turns (even multiplicity) at each zero.
• [ ] State the domain of a rational function by excluding the zeros of the denominator, and distinguish vertical asymptotes (denominator zeros that don't cancel) from holes.
• [ ] Find horizontal asymptotes by comparing degrees: deg numerator < deg denominator → y = 0; equal degrees → y = ratio of leading coefficients; deg numerator > deg denominator → no horizontal asymptote.

What's due this week, and when

Work these in order — each one gets you ready for the next.

Heads-up on the AI tutorial: you'll use a chatbot to draft, and then you judge its work. Chatbots routinely fumble these — they'll say a graph "crosses" at a zero of even multiplicity, or flip the end-behavior rule for a negative leading coefficient. Catching the model is the point.

Late policy reminder: 10% off per day late. If life happens, reach out before the deadline — I'd much rather hear from you early.

How to succeed this week

• Lead with the leading term. For end behavior, all that matters is the degree (odd or even) and the sign of the leading coefficient. Ignore the rest of the polynomial until you need zeros.
• Memorize the touch-vs.-cross rule. Even multiplicity → the graph bounces; odd multiplicity → it punches through. This is the week's most-tested idea.
• For rational functions, factor the denominator first. The zeros of the denominator are where the function is undefined. Then check whether the numerator cancels them (holes) or not (vertical asymptotes).
• Three cases for horizontal asymptotes — say them aloud until automatic. Degree of numerator less than denominator: y = 0. Degrees equal: y = ratio of leading coefficients. Degree of numerator greater: no horizontal asymptote.
• Use Desmos as a reality check. Graph the function and the asymptotes separately; the picture should match your algebra. If they don't match, the algebra found a mistake — not the graph.

You don't need calculus to read a polynomial's shape — just the factored form and two simple rules. See you Tuesday.

(B) Welcome Announcement — Module 10

Release setting: post on the module's start day (offset = 0 days), i.e., Tue Nov 3, 2026 — not before. If your platform won't preserve the scheduled date on import, post this as a draft labeled "Release: Tue Nov 3."

Subject: Week 10 — reading the shape of a function without plotting a point 📐

Hi everyone,

Quick question: have you ever seen an average-cost curve that drops fast at first and then levels off forever, never quite hitting the per-unit cost? Or a drug concentration that spikes and decays toward zero but never quite reaches it? Those are rational functions in the wild — and this week you learn to read them from the algebra alone.

This week — Polynomial & Rational Functions — the big question is: What can the leading term, the zeros, and the asymptotes tell you about the entire shape of a graph? By Friday you'll look at a polynomial in factored form and know immediately which way the tails point, where the graph crosses the x-axis, and where it only touches and turns back — all without plotting a single point. For rational functions you'll find the domain, vertical asymptotes, and horizontal asymptote straight from the algebra.

Three things not to miss:
1. Lecture Tutorial 10 — work through end behavior, multiplicity, and asymptotes with one approved chatbot (Gemini, Claude, or ChatGPT) and submit the share link. You'll catch the model's mistakes, not just trust it. Due Sun Nov 8.
2. Quiz 10 (no AI on quizzes) and Discussion 10 — "Asymptotes in the Wild" also close Sun Nov 8 — the discussion is a quick AI dialogue you summarize and post, so start early and leave time to reply to classmates.
3. Assignment 10 — "Shape Without Plotting" — four AI-coached problems covering end behavior, zeros, domain, asymptotes, and a real-world average-cost model; due Sun Nov 8.

One heads-up: the three horizontal-asymptote cases (less than, equal, greater) are the week's most-tested idea, and chatbots frequently get the "numerator degree bigger" case wrong. Your job, as always: the tool drafts, you judge.

Open the Start Here / Module Overview page first — it lays out everything in order. Bring your factoring skills from Week 6 and your graphing instincts from Week 9. See you Tuesday.

See you soon,
Prof. Calloway