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Mom's Basement University

Polynomial Graphs and End Behavior

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to determine the end behavior of polynomial functions based on their degree and leading coefficient, and how to sketch their graphs using key features. You worked with polynomials in Algebra 2, but here we go deeper into graph analysis and long-term behavior.

The Concept

Section titled “The Concept”

End behavior describes what happens to the graph of a polynomial as x heads toward positive or negative infinity. It’s controlled by just two things: the degree and the sign of the leading coefficient.

The rules:

  • Even degree, positive leading coefficient: both ends go up (like y=x2y = x^2)
  • Even degree, negative leading coefficient: both ends go down (like y=−x2y = -x^2)
  • Odd degree, positive leading coefficient: left end down, right end up (like y=x3y = x^3)
  • Odd degree, negative leading coefficient: left end up, right end down (like y=−x3y = -x^3)

Other key features for sketching:

  • The degree tells you the maximum number of turning points (at most degree - 1)
  • x-intercepts (roots) show where the graph crosses or touches the x-axis
  • Multiplicity of a root matters: odd multiplicity means the graph crosses through, even multiplicity means it touches and bounces back
  • The y-intercept is f(0), which is just the constant term

Worked Example

Section titled “Worked Example”

Example 1: Analyze f(x) = -2x³ + 5x² - 3x + 1

Degree = 3 (odd), leading coefficient = -2 (negative).

End behavior: left end goes up, right end goes down.

y-intercept: f(0)=1\text{y-intercept: } f(0) = 1

Maximum turning points: 3 - 1 = 2.

The graph starts high on the left, has up to 2 turns, and ends going down on the right.

Example 2: Analyze f(x) = x⁴ - 4x²

Degree = 4 (even), leading coefficient = 1 (positive).

End behavior: both ends go up.

Factor to find roots:

f(x)=x2(x2−4)=x2(x−2)(x+2)f(x) = x^2(x^2 - 4) = x^2(x - 2)(x + 2)

Roots: x = 0 (multiplicity 2), x = 2 (multiplicity 1), x = -2 (multiplicity 1).

At x = 0, the graph touches the x-axis and bounces back (even multiplicity). At x = 2 and x = -2, it crosses through (odd multiplicity).

Maximum turning points: 4 - 1 = 3.

Example 3: Determine end behavior of g(x) = 5x⁶ - x³ + 2

Degree = 6 (even), leading coefficient = 5 (positive).

As x→−∞,  g(x)→+∞andas x→+∞,  g(x)→+∞\text{As } x \to -\infty, \; g(x) \to +\infty \quad \text{and} \quad \text{as } x \to +\infty, \; g(x) \to +\infty

Both ends go up. The middle of the graph can have up to 5 turning points.

Real-World Application

Section titled “Real-World Application”

Polynomial graphs and end behavior help model:

  • Business profit functions (revenue minus cost as production scales)
  • Population dynamics over time
  • Engineering stress-strain curves
  • Roller coaster design (polynomial curves with specific turning points)
  • Data fitting in science (polynomial regression)

Example: a company’s profit as a function of units produced might be modeled by a cubic polynomial. The end behavior tells you whether profit eventually grows without bound or eventually turns negative as production increases too far.

Quiz

Section titled “Quiz”
A polynomial with even degree and positive leading coefficient has:
The maximum number of turning points for a degree 5 polynomial is:
For $f(x) = -3x^4 + 2x^3 - 5$, the end behavior is:
If a root has even multiplicity, the graph:
A polynomial with odd degree and negative leading coefficient has: