# 5.6 Rational functions  (Page 2/16)

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## Vertical asymptote

A vertical asymptote    of a graph is a vertical line $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ where the graph tends toward positive or negative infinity as the inputs approach $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ We write

## End behavior of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}$

As the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approach infinity, the function values approach 0. As the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approach negative infinity, the function values approach 0. See [link] . Symbolically, using arrow notation

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 $\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See [link] .

## Horizontal asymptote

A horizontal asymptote    of a graph is a horizontal line $\text{\hspace{0.17em}}y=b\text{\hspace{0.17em}}$ where the graph approaches the line as the inputs increase or decrease without bound. We write

## Using arrow notation

Use arrow notation to describe the end behavior and local behavior of the function graphed in [link] .

Notice that the graph is showing a vertical asymptote at $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ which tells us that the function is undefined at $\text{\hspace{0.17em}}x=2.$

And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at $\text{\hspace{0.17em}}y=4.\text{\hspace{0.17em}}$ As the inputs increase without bound, the graph levels off at 4.

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

End behavior: as Local behavior: as (there are no x - or y -intercepts)

## 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.

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

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

or equivalently, by giving the terms a common denominator,

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

The graph of the shifted function is displayed in [link] .

Notice that this function is undefined at $\text{\hspace{0.17em}}x=-2,\text{\hspace{0.17em}}$ and the graph also is showing a vertical asymptote at $\text{\hspace{0.17em}}x=-2.$

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 $\text{\hspace{0.17em}}y=3.$

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.

The function and the asymptotes are shifted 3 units right and 4 units down. As $\text{\hspace{0.17em}}x\to 3,f\left(x\right)\to \infty ,\text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to ±\infty ,f\left(x\right)\to -4.$

The function is $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{\left(x-3\right)}^{2}}-4.$

## Solving applied problems involving rational functions

In [link] , we shifted a toolkit function in a way that resulted in the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{3x+7}{x+2}.\text{\hspace{0.17em}}$ This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.

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