AP PrecalculusUnit 1A - Polynomial and Rational Functions

1.6 Polynomial Functions and End Behavior

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The end behavior of polynomial functions is determined by their degree.
1.6 Polynomial Functions and End Behavior
1.6 Polynomial Functions and End Behavior
Polynomial Functions and End Behavior

Polynomial Functions and End Behavior

The end behavior of polynomial functions is determined by their degree (highest power of x) and the sign of their leading coefficient (the coefficient of the term with the highest power). This behavior describes how the function behaves as x approaches positive or negative infinity.

Examples

Below are ten examples illustrating different end behaviors based on the degree and leading coefficient:

  1. Function: f(x) = x^3. As x -> ∞, f(x) -> ∞; as x -> -∞, f(x) -> -∞.
  2. Function: f(x) = -2x^4. As x -> ∞ or x -> -∞, f(x) -> -∞.
  3. Function: f(x) = 3x^2 - x. As x -> ∞ or x -> -∞, f(x) -> ∞.
  4. Function: f(x) = x^5 - 4x^3 + 2x. As x -> ∞, f(x) -> ∞; as x -> -∞, f(x) -> -∞.
  5. Function: f(x) = -x^6 + 5x^4 - 3x^2 + 2. As x -> ∞ or x -> -∞, f(x) -> -∞.
  6. Function: f(x) = 4x^3 + 3x^2 - 2x + 1. As x -> ∞, f(x) -> ∞; as x -> -∞, f(x) -> -∞.
  7. Function: f(x) = -0.5x^5 + 100. As x -> ∞, f(x) -> -∞; as x -> -∞, f(x) -> ∞.
  8. Function: f(x) = 2x^7 - 7x^5 + 4. As x -> ∞, f(x) -> ∞; as x -> -∞, f(x) -> -∞.
  9. Function: f(x) = -3x^2 + 6x - 9. As x -> ∞ or x -> -∞, f(x) -> -∞.
  10. Function: f(x) = x^8 + 4x^4 - 2. As x -

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AP Learning Objectives: 
​1.6.A  Describe end behaviors of polynomial functions.

Frequently Asked Questions: End Behavior of Polynomial Functions

What is the end behavior of a polynomial function?
The end behavior of a polynomial function describes how the graph of the function behaves as the input variable \(x\) approaches positive infinity \((x \to \infty)\) or negative infinity \((x \to -\infty)\). It essentially tells you which direction the ends of the graph point.
How do you determine the end behavior of a polynomial function?
The end behavior of a polynomial function is determined by its **leading term** (the term with the highest exponent) when the polynomial is written in standard form (\(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0\)). Specifically, you look at two things:
  1. The **degree** of the polynomial (the highest exponent, \(n\)).
  2. The **leading coefficient** (the coefficient of the term with the highest exponent, \(a_n\)).
The behavior of the leading term as \(x\) goes to \(\pm\infty\) dictates the end behavior of the entire polynomial function.
What are the rules for determining end behavior based on degree and leading coefficient?
Here are the rules based on the degree (\(n\)) and leading coefficient (\(a_n\)):
  • If the degree \(n\) is EVEN: The ends of the graph point in the **same direction**.
    • If \(a_n > 0\) (positive leading coefficient), both ends point **UP** (as \(x \to \pm\infty\), \(f(x) \to \infty\)).
    • If \(a_n < 0\) (negative leading coefficient), both ends point **DOWN** (as \(x \to \pm\infty\), \(f(x) \to -\infty\)).
  • If the degree \(n\) is ODD: The ends of the graph point in **opposite directions**.
    • If \(a_n > 0\) (positive leading coefficient), the left end points **DOWN** (as \(x \to -\infty\), \(f(x) \to -\infty\)) and the right end points **UP** (as \(x \to \infty\), \(f(x) \to \infty\)).
    • If \(a_n < 0\) (negative leading coefficient), the left end points **UP** (as \(x \to -\infty\), \(f(x) \to \infty\)) and the right end points **DOWN** (as \(x \to \infty\), \(f(x) \to -\infty\)).
Why does only the leading term determine the end behavior?
As \(x\) becomes very large (either positively or negatively), the term with the highest exponent (\(a_n x^n\)) grows much faster in magnitude than all the other terms combined. For instance, in \(f(x) = x^3 + 100x\), when \(x=1000\), \(x^3\) is \(1,000,000,000\) while \(100x\) is only \(100,000\). The \(x^3\) term dominates the function's value. Therefore, the behavior of the leading term as \(x \to \pm\infty\) overwhelms the influence of all lower-degree terms, determining the overall direction of the graph's ends.
Does the end behavior tell you about the entire graph of the polynomial?
No, the end behavior only describes what happens at the extreme left and right sides of the graph (as \(x \to \pm\infty\)). It does **not** tell you about the local behavior of the graph in the middle, such as the location of intercepts, turning points (local maxima/minima), or specific shapes between the ends. Understanding local behavior requires analyzing the zeros, using calculus (derivatives), or plotting points.
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