KKhaemenes PrecalculusOpen-Age · Printable Analytical Lesson
Unit 02Course HomeTeacher Key

Unit 02 · Lesson 03 · Week 4 · Polynomial & Rational Functions

Polynomial Graphs from Structure

Connect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. In this lesson, the central focus is polynomial graphs from structure. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

  • Interpret the defining quantities and notation used in polynomial graphs from structure.
  • Construct or analyze at least two representations related to polynomial graphs from structure.
  • Show symbolic work with correct domains, units, intervals, or angle measures.
  • Explain why the selected method is valid and check the result against another representation.

Essential question

How do algebraic structure, zeros, multiplicity, asymptotes, and rates of change determine polynomial and rational behavior?

Multiple-representation model

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy

Connect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure.

f(x)=a_nx^n+\cdots+a_0f(c)=0\Longleftrightarrow (x-c)\text{ is a factor}f(x)=q(x)d(x)+r(x)

Worked reasoning

Structure to graphFactor first to identify zeros and multiplicities, then combine end behavior with selected points.
Domain auditRecord restrictions from the original denominator before simplifying so holes and asymptotes are classified correctly.

Misconception check: A zero with even multiplicity touches the axis; a zero with odd multiplicity crosses it.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Polynomial Graphs from Structure

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U02 · L03
Name
Date
1

State the end behavior of f(x)=−3x⁴+2x.

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
2

Find the zeros of (x−2)²(x+5).

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
3

How does the graph behave at x=2 for (x−2)²(x+5)?

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
4

Divide x³−4x²+x+6 by x−2.

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
5

Find the remainder when f(x)=x³+2x−1 is divided by x−3.

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
6

What is the maximum possible number of turning points of a degree-6 polynomial?

Polynomial visualConnect degree, leading coefficient, zeros, factors, multiplicity, finite differences, and end behavior to polynomial structure. xy
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