Asymptote

Graph of y=1/(x-1)+1 showing an asymptote at x=1
Figure 1: Graph of y=1/(x-1)+1

An asymptote is a linear boundary that a curve gets very close to, but never reaches. Figure 1 is the graph of the function . The vertical asymptote is the line . Notice that is undefined at . This means that gets very close to 0, but never reaches 0. So is an asymptote of .

also has a horizontal asymptote. f(x) can never be 1 because there is no value of x such that is defined. The horizontal asymptote of is .

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Graph of f(x) = 1/(x+1)
Figure 2: Graph of f(x)=1/(x+1)

Figure 2 is a graph of the function . What are the asymptotes of this function?

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Discovery

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Manipulative 1: Graph of f(x)=1/(x-x1)+y1. Created with GeoGebra
 Click on the points on the slider bars in the lower right corner of the manipulative. As you slide the bars, the equation for the function changes. This also changes the asymptotes. What is the relationship between the function and the asymptotes?

More Information

  • asymptote. Dictionary.com. Dictionary.com Unabridged (v 1.1). Random House, Inc. http://dictionary.reference.com/browse/asymptote.
  • Weisstein, Eric W. Asymptote. MathWorld. Wolfram Research. http://mathworld.wolfram.com/Asymptote.html.
  • Asymptote. Wikipedia Encyclopedia. Wikipedia.org http://en.wikipedia.org/wiki/Asymptote.

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