End Behavior of Polynomial Functions (College Board AP® Precalculus): Study Guide

End behavior of polynomial functions

What is end behavior?

• The end behavior of a function describes

  • what happens to the output values

  • as the input values become very large (in the positive or negative direction)

• For a polynomial function $p$ this describes what happens to $p\left(x\right)$

  • as $x$ increases without bound ($x\to \infty$)

  • or decreases without bound ($x\to -\infty$)

• For a nonconstant polynomial function, the output values will always

  • either increase without bound (go to $\infty$)

  • or decrease without bound (go to $-\infty$)

  • in each direction

• Polynomial functions never "level off" to a horizontal asymptote

  • they always increase or decrease without limit

How is end behavior expressed in limit notation?

• End behavior is described using limit notation:

  • $\underset{x\to \infty }{\lim }p\left(x\right)=\infty$ means "as $x$ increases without bound, $p\left(x\right)$ increases without bound"

  • $\underset{x\to \infty }{\lim }p\left(x\right)=-\infty$ means "as $x$ increases without bound, $p\left(x\right)$ decreases without bound"

  • $\underset{x\to -\infty }{\lim }p\left(x\right)=\infty$ means "as $x$ decreases without bound, $p\left(x\right)$ increases without bound"

  • $\underset{x\to -\infty }{\lim }p\left(x\right)=-\infty$ means "as $x$ decreases without bound, $p\left(x\right)$ decreases without bound"

• A complete description of a polynomial's end behavior includes both limits

  • i.e. what happens as $x\to \infty$ and as $x\to -\infty$

How do I determine the end behavior of a polynomial?

• First, identify the leading term

  • i.e. the term with the highest power of $x$

  • Remember that the leading term might not be the first term written in the expression (see the Polynomial Functions study guide)

• Then use the degree of the polynomial (odd or even) and the sign of the leading coefficient (positive or negative) to determine the end behavior

Why does the leading term determine end behavior?

• As the input values increase or decrease without bound

  • the values of the leading term dominate the values of all lower-degree terms

  • This is because higher powers of $x$ grow much faster than lower powers

• E.g. for $p\left(x\right)=2x^4-50x^3+100x$

  • when $x$ is very large, the $2x^4$ term is far larger in magnitude than the $-50x^3$ and $100x$ terms

• Therefore, the degree of the polynomial and the sign of the leading coefficient are all you need to determine the end behavior

How can I remember the four cases?

• For odd degree polynomials, the two ends behave in opposite ways:

  • Positive leading coefficient → falls left, rises right (think of $x^3$)

  • Negative leading coefficient → rises left, falls right (think of $-x^3$)

• For even degree polynomials, both ends behave the same way

  • Positive leading coefficient → both ends go up (think of $x^2$)

  • Negative leading coefficient → both ends go down (think of $-x^2$)

• For the negative coefficient cases

  • note that multiplying by a negative number flips the end behavior compared to the positive coefficient case