Precalculus and Trigonometry · 40 minutes
Suggested grades 11–12 · Unit 1: Functions, transformations and rational graphs · Lesson 2 of 2
Polynomial Zeros and Rational Graph Boundaries
Use factors and a data table to distinguish roots, exclusions and graph behavior.
Mr. S Teaches · Original starter lesson · HS-PRE-02 · Grade range is planning guidance.Watch, notice, discuss
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Zeros of polynomials introduction | Polynomial graphs | Algebra 2 | Khan Academy
By Khan Academy
Connect factors, zeros and x-intercepts before sketching the polynomial.
Open on YouTube: Zeros of polynomials introduction | Polynomial graphs | Algebra 2 | Khan Academy (opens a new tab) ↗Graphing Rational Functions With Vertical, Horizontal & Slant Asymptotes, Holes, Domain & Range
By The Organic Chemistry Tutor
Choose relevant rational-graph examples and compare excluded inputs, holes and asymptotes; no calculus is needed.
Open on YouTube: Graphing Rational Functions With Vertical, Horizontal & Slant Asymptotes, Holes, Domain & Range (opens a new tab) ↗How are a zero, excluded input, hole and vertical asymptote different?
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Learning objective
Sketch a polynomial and a rational function with correctly labeled zeros and excluded inputs.
Before you begin
- Factoring
- Polynomial division
- Coordinate graphing
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- zero
- An allowed input where a function’s output is zero.
- vertical asymptote
- A vertical line near which a graph’s values grow in magnitude without bound.
- removable hole
- A missing graph point caused by an excluded input after cancellation.
Understand the concept
Factoring can reveal polynomial zeros and how signs change across them. Rational graphs require additional checks because denominator zeros are excluded. A canceled factor may leave a hole rather than a vertical asymptote. Numerical samples help sketch behavior, but the algebraic domain is the basis for exclusions.
Teach it step by step
- Factor p(x)=x²−x−6, locate zeros−2 and 3 and test signs at−3,0,4.
- Rewrite r(x)=(x²−1)/(x−1) as x+1 for x≠1 and mark its hole at(1,2).
- Inspect q(x)=1/(x−2) at x=1,1.9,1.99,2.01,2.1,3 to sketch its two branches.
- Compare q at large positive and negative inputs and mark its horizontal asymptote y=0 without assigning forbidden values.
Make the thinking visible
Worked example or model response
p(x)=(x−3)(x+2), so the upward-opening graph crosses at−2 and 3. r follows y=x+1 except the missing point(1,2). q has no zero, excludes 2, and has asymptotes x=2 and y=0.
Guided practice
Sketch y=(x−1)(x+4) using zeros, sign samples and the positive leading term. Identify the hole in(x²−9)/(x−3).
Independent practice
1. State zeros and vertical asymptote of(x+1)/(x−4). 2. Rewrite it as 1+5/(x−4) and describe its horizontal asymptote.
Quick assessment
Why is x=1 not a zero of r(x), despite being a numerator zero in its original form?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: Zeros 1 and−4, upward opening; the rational expression has a hole at(3,6).
- Independent 1: Zero−1; vertical asymptote x=4. 2: Horizontal asymptote y=1 because the extra fraction becomes small for large input magnitude.
- Quick check: x=1 is excluded by the denominator, so no function value exists there.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use separate lists for numerator zeros, denominator exclusions and canceled factors.
Extension challenge
Construct a rational function with one hole and one distinct vertical asymptote.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.IF.C.7.c
Common Core · High school · Functions
Regular high school mathematics standard
Use factorable polynomial zeros and sign samples to sketch a graph.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.
Contributes to Common Core standard
HSF.IF.C.7.d
Common Core · High school · Functions
(+) Additional preparation for advanced study
Sketch rational functions with zeros, holes, asymptotes and end behavior.
Read the official standard ↗Additional preparation (+). These simple factorable examples contribute rational graphing practice; more complicated cases need technology and further work.
Verified 2026-11-04. Selected contribution, not full attainment.