Polynomial End Behavior

Predict how a function behaves as x approaches infinity using the Leading Term.

Description:

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Limit Notation:

Leading Coefficient Test

The End Behavior of a polynomial function is determined entirely by its leading term (the term with the highest exponent). As x gets extremely large (positive) or extremely small (negative), the leading term dominates the value of the function.

The Four Cases

We look at two things: whether the Degree (n) is Even or Odd, and whether the Leading Coefficient (aâ‚™) is Positive or Negative.

Degree	Leading Coeff +	Leading Coeff -
Odd	Falls Left, Rises Right â†™ ... â†—	Rises Left, Falls Right â†– ... â†˜
Even	Rises Left, Rises Right â†– ... â†—	Falls Left, Falls Right â†™ ... â†˜

Why it Works

Consider f(x) = xÂ³ - 1000xÂ². Even though -1000xÂ² seems huge, if you plug in x = 1,000,000, the xÂ³ term (10Â¹â¸) is vastly larger than the square term. Thus, the sign of the xÂ³ term dictates whether the function goes to positive or negative infinity.