End Behavior and Local Behavior of Rational Functions

As the values of $x$ approach infinity, the function values approach $0$. As the values of $x$ approach negative infinity, the function values approach $0$. See Figure 4. Symbolically, using arrow notation

$\text { As } x \rightarrow \infty, f(x) \rightarrow 0 \text {, and as } x \rightarrow-\infty, f(x) \rightarrow 0$

Figure 4

Based on this overall behavior and the graph, we can see that the function approaches $0$ but never actually reaches $0$; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line $y=0$. See Figure 5.

Figure 5

HORIZONTAL ASYMPTOTE

A horizontal asymptote of a graph is a horizontal line $y=b$ where the graph approaches the line as the inputs increase or decrease without bound. We write

$\text { As } x \rightarrow \infty \quad \text { or } x \rightarrow-\infty, \quad f(x) \rightarrow b$

EXAMPLE 1

Using Arrow Notation

Use arrow notation to describe the end behavior and local behavior of the function graphed in Figure 6.

Figure 6

Solution

Notice that the graph is showing a vertical asymptote at $x=2$, which tells us that the function is undefined at $x=2$.

$\text { As } x \rightarrow 2^­, f(x) \rightarrow-\infty, \quad \text { and as } x \rightarrow 2^{+}, \quad f(x) \rightarrow \infty$.

And as the inputs decrease without bound, the graph appears to be leveling off at output values of $4$, indicating a horizontal asymptote at $y=4$. As the inputs increase without bound, the graph levels off at $4$.

$\text { As } x \rightarrow \infty, \quad f(x) \rightarrow 4 \text { and as } x \rightarrow-\infty, \quad f(x) \rightarrow 4$.

TRY IT #1

Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.

EXAMPLE 2

Using Transformations to Graph a Rational Function

Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.

Solution

Shifting the graph left $2$ and up $3$ would result in the function

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

or equivalently, by giving the terms a common denominator,

$f(x)=\frac{3 x+7}{x+2}$

The graph of the shifted function is displayed in Figure 7.

Figure 7

Notice that this function is undefined at $x=−2$, and the graph also is showing a vertical asymptote at $x=−2$.

$\text { As } x \rightarrow-2^­, f(x) \rightarrow-\infty \text {, and as } x \rightarrow-2^{+}, f(x) \rightarrow \infty$.

As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of $3$, indicating a horizontal asymptote at $y=3$.

$\text { As } x \rightarrow \pm \infty, f(x) \rightarrow 3$

Analysis

Notice that horizontal and vertical asymptotes are shifted left $2$ and up $3$ along with the function.

TRY IT #2

Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right $3$ units and down $4$ units.