End behavior

Definition

End behavior describes the behavior of the graph of a function as the input values approach positive or negative infinity. It helps to understand how the function behaves at the extremes of its domain.

5 Must Know Facts For Your Next Test

1. The end behavior of polynomial functions is determined by their leading term.
2. For a polynomial function $f(x) = ax^n + ...$, if $a > 0$ and $n$ is even, both ends of the graph will go upwards.
3. If $a > 0$ and $n$ is odd, the left end will go downwards and the right end will go upwards.
4. Rational functions' end behavior can be found using their horizontal asymptotes or by analyzing their highest-degree terms in numerator and denominator.
5. Asymptotic behavior, such as vertical asymptotes, often affects but does not solely determine end behavior.

Review Questions

• How does the leading coefficient and degree of a polynomial influence its end behavior?
• What determines the horizontal asymptote of a rational function?
• In which cases would you consider vertical asymptotes when discussing end behavior?

Related terms

Leading Term: The term in a polynomial with the highest power; it determines the end behavior of the polynomial.

Horizontal Asymptote: A horizontal line that represents where a function's value approaches as x approaches infinity or negative infinity.

Vertical Asymptote: $x$-values where a function tends towards positive or negative infinity.