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: if the numerator’s degree is lower, the graph approaches 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.

To check whether the graph crosses that line somewhere in the middle, set f(x) equal to the line’s equation and solve; crossing in the middle does not change end behavior. This tells you 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. \(\color{blue}{y=x-3}\)

End Behavior of Rational Functions: Summary Table

End behavior describes what f(x) does as x → ∞ (far right) and x → −∞ (far left). Use the degrees to pick the right row.

A fast check for the last two rows: divide the leading terms. For (−x³ + 4)/(x − 2), that is −x³/x = −x². A downward parabola falls on both ends, so f(x) does too.

Worked Example: Slant Asymptote and Vertical Asymptote

Find the end behavior of f(x) = (2x² + 1)/(x − 1).

1. Compare degrees: the numerator has degree 2 and the denominator has degree 1, so expect a slant asymptote.
2. Divide. 2x² ÷ x = 2x, and 2x(x − 1) = 2x² − 2x. Subtract to get 2x + 1. Then 2x ÷ x = 2, and 2(x − 1) = 2x − 2. Subtract to get a remainder of 3.
3. So f(x) = 2x + 2 + 3/(x − 1). As x gets very large or very small, 3/(x − 1) gets close to 0, and the graph hugs y = 2x + 2.
4. End behavior: as x → ∞, f(x) → ∞; as x → −∞, f(x) → −∞.

End behavior only tells you about the far left and far right. To find the vertical asymptote, set the denominator equal to 0: x − 1 = 0, so x = 1. The numerator at x = 1 is 3, not 0, so x = 1 is a true vertical asymptote.

Frequently Asked Questions

How do you find the end behavior of a rational function?

Compare the degree of the numerator, n, with the degree of the denominator, m. If n < m, the graph levels off toward y = 0. If n = m, it approaches the ratio of the leading coefficients. If n > m, divide to find the line or curve the graph follows. For f(x) = (5x + 2)/(x² − 1), the bottom degree is larger, so f(x) → 0 as x → ±∞.

How do you find the end behavior of a rational function without graphing?

Look only at the leading terms, since they dominate when x is huge. For (8x³ − x + 4)/(2x³ + 9x²), the leading terms give 8x³/2x³ = 4, so the graph approaches y = 4 at both ends. If the ratio of leading terms still contains x, like 3x²/x = 3x, the function grows without bound, and long division gives the exact asymptote.

How do you find the end behavior asymptote of a rational function?

The end behavior asymptote, or EBA, is the simplest function the graph hugs as x → ±∞. Divide the numerator by the denominator and keep the quotient, dropping the remainder. For (x² + 3x + 5)/(x + 1), division gives x + 2 with remainder 3, so the EBA is y = x + 2. When the degrees are equal or the bottom degree is bigger, the EBA is a horizontal line.

How do you find the end behavior of a slant asymptote?

A slant asymptote appears when the numerator’s degree is exactly one more than the denominator’s, and the graph runs alongside that line at both ends. For (2x² − x)/(x − 3), long division gives 2x + 5 with remainder 15, so the asymptote is y = 2x + 5. As x → ∞ the graph rises with the line, and as x → −∞ it falls with it.

What are some examples of end behavior of rational functions?

Three quick ones cover every case. (x − 4)/(x² + 2) approaches y = 0 because the bottom degree is larger. (−6x² + x)/(2x² + 7) approaches y = −3, the ratio −6/2. (x³ + 1)/(x − 2) has a numerator two degrees higher, so it behaves like the quotient x² + 2x + 4 and rises toward +∞ on both the left and the right.