How to Find the End Behavior of Rational Functions?

The end behavior of a rational function is decided by comparing the degrees of numerator and denominator: lower gives y = 0, equal gives the ratio of leading coefficients, and higher by one gives a slant asymptote. Below: all three cases with worked examples.

A rational function is a function that consists of a ratio of polynomials. Rational functions are of this form \(f(x)=\frac {q(x)}{p(x)}\), where \(q(x)\) and \(p(x)\) are polynomials and \(p(x) ≠0\).

End Behavior: The end behavior of a function \(f(x)\) describes the behavior of the function when \(x→ +∞\) or \(x→ -∞\). The end behavior of a function is equal to the horizontal asymptotes, slant/oblique asymptotes, or the quotient obtained when long dividing the polynomials.

Related Topics

How to Graph Rational Functions

A step-by-step guide to the end behavior of rational functions

A rational function’s final behavior can take one of three forms:

• Examine the numerator and denominator degrees. There is a horizontal asymptote of \(y=0\) if the degree of the denominator is greater than the degrees of the numerator, which is the function’s end behavior.

• If the degree of the numerator and denominator are identical, there is a horizontal asymptote of \(y=\frac{a}{b}\), where \(a\) is the numerator’s leading coefficient and \(b\) is the denominator’s leading coefficient. This is the final result.

• There is a slant/oblique asymptote (if the numerator’s degree is precisely one bigger than the denominator’s degree) or the function is asymptotic to a polynomial if the numerator’s degree is more than the denominator’s degree. To get the quotient, long division of the polynomials in each case (what you have without the remainder). Then \(y=g(x)\), where \(g(x)\) is the quotient that determines the outcome.

It is necessary to solve the system of equations that consists of the equation of the end behavior and the rational function to determine if the function touches or crosses its asymptote to make this decision.

The End Behavior of Rational Functions, Example 1:

Find the end behavior of the rational function. \(f(x)=\frac{6x^3+3x^2-x-2}{3x^3+5x-4}\)

The degrees of the numerator and denominator are equal. So we have a horizontal asymptote of:

\(y=\frac{6}{3}=2\) → \(y=2\)

The end behavior of the rational function is the horizontal asymptote \(y=2\). It means that the graph of the function \(f(x)=\frac{6x^3+3x^2-x-2}{3x^3+5x-4}\) approach the graph of the horizontal line \(y=2\) when \(x→ +∞\) and \(x→ -∞\).

The End Behavior of Rational Functions, Example 2:

Find the end behavior of the rational function. \(f(x)=\frac{x^2-4}{x^2-9}\)

The degrees of the numerator and denominator are equal. So we have a horizontal asymptote of:

\(y=\frac{1}{1}=1\) → \(y=1\)

The end behavior of the rational function is the horizontal asymptote \(y=1\).

Exercises for the End Behavior of Rational Functions

Find the end behavior of the rational function.

1. \(\color{blue}{f(x)=\frac{1}{x-3}+3}\)
2. \(\color{blue}{f(x)=\frac{x^2+2x-3}{-3x-6}}\)
3. \(\color{blue}{f(x)=\frac{x^4+3x^2-1}{2x^3+5x}}\)
4. \(\color{blue}{f(x)=\frac{2x-5}{x^2+3}}\)
5. \(\color{blue}{f(x)=\frac{x^3-3x+2}{x^2+3x-10}}\)

1. \(\color{blue}{y=3}\)
2. \(\color{blue}{y=-\frac{1}{3}x}\)
3. \(\color{blue}{y=\frac{1}{2}x}\)
4. \(\color{blue}{y=0}\)
5. \(\color{blue}{y=x-3}\)