End Behavior of Polynomial Functions

Core Concepts

Learn to predict the end behavior of a polynomial's graph by examining its degree and the sign of its leading coefficient.

End Behavior of Polynomial Functions

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What does the 'end behavior' of a polynomial function's graph describe?

It describes the direction of the graph at its far left and far right ends. In other words, it describes the value of \(f(x)\) as \(x\) approaches positive infinity (\(x \to \infty\)) and as \(x\) approaches negative infinity (\(x \to -\infty\)).

What two features of a polynomial determine its end behavior?

1. The degree of the polynomial (whether it's even or odd). 2. The sign of the leading coefficient (whether it's positive or negative).

Describe the end behavior of a polynomial with an EVEN degree and a NEGATIVE leading coefficient.

A downward-opening parabola on a Cartesian plane, demonstrating the end behavior of an even-degree polynomial with a negative leading coefficient, falling toward negative infinity in both directions.

The graph falls to the left and falls to the right. As \(x \to \infty\), \(f(x) \to -\infty\). As \(x \to -\infty\), \(f(x) \to -\infty\). (Think of \(y=-x^2\))

Describe the end behavior of a polynomial with an ODD degree and a NEGATIVE leading coefficient.

Cartesian graph of y equals negative x cubed, illustrating the end behavior of an odd-degree polynomial with a negative leading coefficient rising to the left and falling to the right.

The graph rises to the left and falls to the right. As \(x \to \infty\), \(f(x) \to -\infty\). As \(x \to -\infty\), \(f(x) \to \infty\). (Think of \(y=-x^3\))

Determine the end behavior of \(f(x) = -3x^5 + 2x^2 - 7\).

The degree is \(5\) (odd) and the leading coefficient is \(-3\) (negative). Therefore, the graph rises to the left and falls to the right. As \(x \to -\infty\), \(f(x) \to \infty\) and as \(x \to \infty\), \(f(x) \to -\infty\).

Determine the end behavior of \(P(x) = x^2(5 - 2x^4)\).

First, find the leading term by multiplying: \(x^2(-2x^4) = -2x^6\). The degree is \(6\) (even) and the leading coefficient is \(-2\) (negative). Therefore, the graph falls to the left and falls to the right. As \(x \to \pm\infty\), \(f(x) \to -\infty\).

Describe the end behavior of a polynomial with an EVEN degree and a POSITIVE leading coefficient.

Graph of an upward-opening parabola on a coordinate plane, illustrating end behavior that rises to positive infinity on both the left and right sides.

The graph rises to the left and rises to the right. As \(x \to \infty\), \(f(x) \to \infty\). As \(x \to -\infty\), \(f(x) \to \infty\). (Think of \(y=x^2\))

Describe the end behavior of a polynomial with an ODD degree and a POSITIVE leading coefficient.

Graph of an odd-degree polynomial with a positive leading coefficient, passing through the origin, falling toward negative infinity to the left, and rising toward positive infinity to the right.

The graph falls to the left and rises to the right. As \(x \to \infty\), \(f(x) \to \infty\). As \(x \to -\infty\), \(f(x) \to -\infty\). (Think of \(y=x^3\))

End Behavior of Polynomial Functions

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