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Graph of f(x)=1/x with its vertical asymptote at x=0.

Vertical asymptote

A vertical asymptote    of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a . We write

As  x a , f ( x ) ,   or as  x a , f ( x ) .

End behavior of f ( x ) = 1 x

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 [link] . Symbolically, using arrow notation

As  x , f ( x ) 0 , and as  x , f ( x ) 0.

Graph of f(x)=1/x which highlights the segments of the turning points to denote their end behavior.

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 [link] .

Graph of f(x)=1/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.

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

As  x  or  x ,   f ( x ) b .

Using arrow notation

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

Graph of f(x)=1/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.

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

As  x 2 , f ( x ) ,  and as  x 2 + ,   f ( x ) .

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.

As  x ,   f ( x ) 4  and as  x ,   f ( x ) 4.
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Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.

End behavior: as x ± ,   f ( x ) 0 ; Local behavior: as x 0 ,   f ( x ) (there are no x - or y -intercepts)

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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 ( x ) = 1 x + 2 + 3

or equivalently, by giving the terms a common denominator,

f ( x ) = 3 x + 7 x + 2

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

Graph of f(x)=1/(x+2)+3 with its vertical asymptote at x=-2 and its horizontal asymptote at y=3.

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

As  x 2 ,   f ( x ) , and as   x 2 + ,   f ( x ) .

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.

As  x ± ,   f ( x ) 3.
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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.

Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.

The function and the asymptotes are shifted 3 units right and 4 units down. As x 3 , f ( x ) , and as x ± , f ( x ) 4.

The function is f ( x ) = 1 ( x 3 ) 2 4.

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Solving applied problems involving rational functions

In [link] , we shifted a toolkit function in a way that resulted in the function f ( x ) = 3 x + 7 x + 2 . 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.

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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