MA001: College Algebra

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial's end behavior will mirror that of the leading term. Likewise, a rational function's end behavior will mirror that of the ratio of the function that is the ratio of the leading terms.

There are three distinct outcomes when checking for horizontal asymptotes:

Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at $y=0$.

$\text { Example: } f(x)=\frac{4 x+2}{x^{2}+4 x-5}$

In this case, the end behavior is $f(x) \approx \frac{4 x}{x^{2}}=\frac{4}{x}$. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function $g(x)=\frac{4}{x}$, and the outputs will approach zero, resulting in a horizontal asymptote at $y=0$. See Figure 12. Note that this graph crosses the horizontal asymptote.

Figure 12 Horizontal asymptote $y=0$ when $f(x)=\frac{p(x)}{q(x)}, \quad q(x) \neq 0$ where degree of $p$ < degree of $q$

Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.

$\text { Example: } f(x)=\frac{3 x^{2}-2 x+1}{x-1}$

In this case, the end behavior is $f(x) \approx \frac{3 x^{2}}{x}=3 x$. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function $g(x)=3x$. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of $g(x)=3x$ looks like a diagonal line, and since $f$ will behave similarly to $g$, it will approach a line close to $y=3x$. This line is a slant asymptote.

To find the equation of the slant asymptote, divide $\frac{3 x^{2}-2 x+1}{x-1}$. The quotient is $3x+1$, and the remainder is $2$. The slant asymptote is the graph of the line $g(x)=3x+1$. See Figure 13.

Figure 13 Slant asymptote when $f(x)=\frac{p(x)}{q(x)}, \quad q(x) \neq 0$ where degree of $p$ > degree of $q$ by $1$

Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at $y=\frac{a_{n}}{b_{n}}$, where $a_{n}$ and $b_{n}$ are the leading coefficients of $p(x)$ and $q(x)$ for $f(x)=\frac{p(x)}{q(x)}, q(x) \neq 0$.

$\text { Example: } f(x)=\frac{3 x^{2}+2}{x^{2}+4 x-5}$

In this case, the end behavior is $f(x) \approx \frac{3 x^{2}}{x^{2}}=3$. This tells us that as the inputs grow large, this function will behave like the function $g(x)=3$, which is a horizontal line. As $x \rightarrow \pm \infty, f(x) \rightarrow 3$, resulting in a horizontal asymptote at $y=3$. See Figure 14. Note that this graph crosses the horizontal asymptote.

Figure 14 Horizontal asymptote when $f(x)=\frac{p(x)}{q(x)}, \quad q(x) \neq 0$ where degree of $p$ = degree of $q$

Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.

It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function

$f(x)=\frac{3 x^{5}-x^{2}}{x+3}$

with end behavior

$f(x) \approx \frac{3 x^{5}}{x}=3 x^{4}$,

the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.

$x \rightarrow \pm \infty, f(x) \rightarrow \infty$

HORIZONTAL ASYMPTOTES OF RATIONAL FUNCTIONS

The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.

1. Degree of numerator is less than degree of denominator: horizontal asymptote at $y=0$.

2. Degree of numerator is greater than degree of denominator by one: no horizontal asymptote; slant asymptote.

3. Degree of numerator is equal to degree of denominator: horizontal asymptote at ratio of leading coefficients.

EXAMPLE 7

Identifying Horizontal and Slant Asymptotes

For the functions listed, identify the horizontal or slant asymptote.

a. $g(x)=\frac{6 x^{3}-10 x}{2 x^{3}+5 x^{2}}$

b. $h(x)=\frac{x^{2}-4 x+1}{x+2}$

c. $k(x)=\frac{x^{2}+4 x}{x^{3}-8}$

Solution

For these solutions, we will use $f(x)=\frac{p(x)}{q(x)}, q(x) \neq 0$.

a. $g(x)=\frac{6 x^{3}-10 x}{2 x^{3}+5 x^{2}}$: The degree of $p=\text { degree of } q=3$, so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at $y=\frac{6}{2}$ or $y=3$.

b. $h(x)=\frac{x^{2}-4 x+1}{x+2}$: The degree of $p=2$ and degree of $q = 1$. Since $p > q$ by $1$, there is a slant asymptote found at $\frac{x^{2}-4 x+1}{x+2}$.

The quotient is $x–6$ and the remainder is $13$. There is a slant asymptote at $y=x–6$.

c. $k(x)=\frac{x^{2}+4 x}{x^{3}-8}$: The degree of $p=2$ < degree of $q=3$, so there is a horizontal asymptote $y=0$.

EXAMPLE 8

Identifying Horizontal Asymptotes

In the sugar concentration problem earlier, we created the equation $C(t)=\frac{5+t}{100+10 t}$.

Solution

Both the numerator and denominator are linear (degree $1$). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is $t$, with coefficient $1$. In the denominator, the leading term is $10t$, with coefficient $10$. The horizontal asymptote will be at the ratio of these values:

$t \rightarrow \infty, C(t) \rightarrow \frac{1}{10}$

This function will have a horizontal asymptote at $y=\frac{1}{10}$.

This tells us that as the values of t increase, the values of $C$ will approach $\frac{1}{10}$. In context, this means that, as more time goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or $\frac{1}{10}$ pounds per gallon.

EXAMPLE 9

Identifying Horizontal and Vertical Asymptotes

Find the horizontal and vertical asymptotes of the function

$f(x)=\frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)}$

Solution

First, note that this function has no common factors, so there are no potential removable discontinuities.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at $x=1,–2$,and $5$, indicating vertical asymptotes at these values.

The numerator has degree $2$, while the denominator has degree $3$. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as $x \rightarrow \pm \infty, f(x) \rightarrow 0$. This function will have a horizontal asymptote at $y=0$. See Figure 15.

Figure 15

TRY IT #6

Find the vertical and horizontal asymptotes of the function:

$f(x)=\frac{(2 x-1)(2 x+1)}{(x-2)(x+3)}$