Week 10 — Polynomial & Rational Functions

Welcome to your Week 10 Start Here page.

Why this matters

A small-business owner reads a horizontal asymptote to see how low the cost per item can go, and a nursing student reads one when a medicine level climbs and then settles.

In plain language

This week you learn to read the shape of a graph from its formula before you plot a single point. A polynomial function is a function built from terms with whole-number powers, like Week 9's parabolas, and you will find where its graph heads and where it hits the x-axis. For a rational function (one polynomial divided by another), you will also find asymptotes, the guide lines a graph creeps toward without arriving. These lines answer real questions, such as how cheap an item can get when a business makes more of them. Every feature you find from the formula, you will confirm on a graph in Desmos or GeoGebra.

• Already covered: Week 3 introduced domain and rational functions; Week 6 multiplied and factored polynomials; Week 7 solved quadratic equations; Week 9 graphed the parabola.
• This week: the graphs of polynomial functions of any degree and of rational functions, read from their formulas.
• How they connect: a zero is a Week 7 solution, a factor is a Week 6 factor, and a domain is a Week 3 domain, now read off a picture.
• Where we're going next: Week 11, Rational Expressions & Equations, simplifies, adds, and solves the fractions whose graphs you meet here.

Focus

What can a formula tell you about a graph's ends, zeros, and asymptotes before you plot a point?

By the end of this module, you should be able to:

• [ ] read a polynomial's end behavior from its leading term
• [ ] find zeros and say whether the graph crosses or touches
• [ ] tell a hole from a vertical asymptote by factoring first
• [ ] find a horizontal asymptote and explain it in a setting

To-do

Plan your week

1. Read Chapter 10 (about 50 minutes), then the lecture and slides.
2. Work the lecture tutorial (60–90 minutes) and the practice set (about 20 minutes).
3. Do the Model/Build Studio (about 90 minutes), the discussion, and the assignment.
4. Take the quiz.

This week's checkpoint: for R(x) = (x + 1) ⁄ (x² − 4), name both asymptote types and both intercepts.

You've got this

Factor first, and ask your instructor early if stuck.

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