Polynomial Graph Behavior

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Describe the end behavior of a polynomial using its leading term.
  • Identify roots and determine whether the graph crosses or touches the x-axis.
  • Understand how degree and multiplicity shape a polynomial’s graph.

Key Ideas

Polynomial graphs follow predictable patterns based on their leading term and root multiplicities.

End Behavior

End behavior depends entirely on the leading term \(ax^n\).

  • If \(n\) is even:
    • \(a > 0\) → up–up
    • \(a < 0\) → down–down
  • If \(n\) is odd:
    • \(a > 0\) → down–up
    • \(a < 0\) → up–down

Roots & Multiplicity

  • A root occurs when \(f(r) = 0\).

  • When written in factored form:
    \[ f(x) = (x - r)^m \]

    • Odd multiplicity (\(m\) odd): graph crosses the x-axis
    • Even multiplicity (\(m\) even): graph touches/bounces off the x-axis

Common Problem Types

1. Determining End Behavior

Look only at the leading term’s degree and coefficient.

2. Identifying Roots From Factored Form

Each factor \((x - r)^m\) gives root \(r\) with multiplicity \(m\).

3. Predicting Crossing vs. Touching

Odd multiplicity → crossing
Even multiplicity → touching/bouncing

4. Mixed Polynomial Expressions

Sometimes you need to expand or recognize the leading term before analyzing.

Strategies

  • Always rewrite the polynomial in descending powers of \(x\) before analyzing.
  • Focus on the leading term for end behavior—ignore everything else.
  • Use multiplicity to decide how the graph behaves at each root.
  • Combine your knowledge: the degree gives overall behavior; multiplicity gives local behavior.
  • If unsure, sketch rough behavior using end behavior + roots + multiplicities.

Worked Examples

Example 1 — Determine End Behavior

Analyze:
\[ f(x) = -3x^4 + 7x - 2 \]

Leading term: \(-3x^4\)

  • Degree: even
  • Leading coefficient: negative

End behavior: down–down

Example 2 — Analyze Multiplicity

Given:
\[ f(x) = (x + 2)^2 (x - 1) \]

Roots:

  • \(x = -2\) (multiplicity 2 → touches the x-axis)
  • \(x = 1\) (multiplicity 1 → crosses the x-axis)
WarningCommon Mistakes
  • Looking at the constant term instead of the leading term for end behavior.
  • Assuming every root crosses the axis—multiplicity matters.
  • Ignoring the sign of the leading coefficient when describing the graph’s direction.

Practice Problems

  1. Determine the end behavior of \(4x^5 - 7x + 3\).
  2. Determine the end behavior of \(-2x^6 + x^2 - 1\).
  3. For \(f(x) = (x - 3)^3\), does the graph cross or touch at \(x = 3\)?
  4. List roots and multiplicities of \((x + 5)^2 (x - 1)^4\).
  5. Determine the end behavior of \((x - 2)^2(x + 1)^3\).

1.
Leading term: \(4x^5\) → odd degree, positive coefficient → down–up

2.
Leading term: \(-2x^6\) → even degree, negative coefficient → down–down

3.
Multiplicity 3 (odd) → crosses

4.
Roots:

  • \(x = -5\) (multiplicity 2)
  • \(x = 1\) (multiplicity 4)

5.
Leading behavior: degree \(5\) (odd), positive leading coefficient → up–down

Summary

  • End behavior depends only on the leading term’s degree and coefficient.
  • Even degree → ends move in the same direction; odd degree → opposite directions.
  • Positive coefficients lift the right side; negative coefficients pull it down.
  • Multiplicity determines whether the graph crosses or only touches the x-axis.
  • Factored form makes it easy to identify roots and multiplicities.
  • Ignore all but the leading term when describing end behavior.
  • Even multiplicity → touch; odd multiplicity → cross.
  • Look for factored form—it reveals roots instantly.
  • Use multiplicity + end behavior to sketch quick polynomial graphs.