# Difference between revisions of "Asymptote (geometry)"

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== Horizontal Asymptotes == | == Horizontal Asymptotes == | ||

For rational functions in the form of <math>\frac{P(x)}{Q(x)}</math> where <math>P(x), Q(x)</math> are both [[polynomial]]s: | For rational functions in the form of <math>\frac{P(x)}{Q(x)}</math> where <math>P(x), Q(x)</math> are both [[polynomial]]s: | ||

+ | |||

1. If the degree of <math>Q(x)</math> is greater than that of the degree of <math>P(x)</math>, then the horizontal asymptote is at <math>y = 0</math>. | 1. If the degree of <math>Q(x)</math> is greater than that of the degree of <math>P(x)</math>, then the horizontal asymptote is at <math>y = 0</math>. | ||

2. If the degree of <math>Q(x)</math> is equal to that of the degree of <math>P(x)</math>, then the horizontal asymptote is at the quotient of the leading coefficient of <math>P(x)</math> over the leading coefficient of <math>Q(x)</math>. | 2. If the degree of <math>Q(x)</math> is equal to that of the degree of <math>P(x)</math>, then the horizontal asymptote is at the quotient of the leading coefficient of <math>P(x)</math> over the leading coefficient of <math>Q(x)</math>. | ||

− | 3. If the degree of <math>Q(x)</math> is less than the degree of <math>P(x), see below (slanted asymptotes) | + | 3. If the degree of <math>Q(x)</math> is less than the degree of <math>P(x)</math>, see below (slanted asymptotes) |

A function may not have more than one horizontal asymptote. Functions with a "middle section" may cross the horizontal asymptote at one point. To find this point, set y=horizontal asymptote and solve. | A function may not have more than one horizontal asymptote. Functions with a "middle section" may cross the horizontal asymptote at one point. To find this point, set y=horizontal asymptote and solve. | ||

===Example Problem=== | ===Example Problem=== | ||

− | Find the horizontal asymptote of < | + | Find the horizontal asymptote of <math>f(x) = \frac{x^2 - 3x + 2}{-2x^2 + 15x + 10000}</math>. |

+ | |||

====Solution==== | ====Solution==== | ||

The numerator has the same degree as the denominator, so the horizontal asymptote is the quotient of the leading coefficients: | The numerator has the same degree as the denominator, so the horizontal asymptote is the quotient of the leading coefficients: | ||

− | < | + | <math>y= \frac {1} {-2}</math> |

== Slanted Asymptotes == | == Slanted Asymptotes == |

## Revision as of 18:13, 27 June 2012

*For the vector graphics language, see Asymptote (Vector Graphics Language).*

An **asymptote** is a line or curve that a certain function approaches.

Linear asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).

## Contents

## Vertical Asymptotes

The vertical asymptote can be found by finding values of that make the function undefined. Generally, it is found by setting the denominator of a rational function to zero.

If the numerator and denominator of a rational function share a factor, this factor is not a vertical asymptote. Instead, it appears as a hole in the graph.

A rational function may have more than one vertical asymptote.

### Example Problems

Find the vertical asymptotes of 1) 2) .

#### Solution

1) To find the vertical asymptotes, let . Solving the equation:

$\begin{eqnarray*}x^2-5x&=&0\\x&=&\boxed{0,5}\end{eqnarray*}$ (Error compiling LaTeX. ! Missing \endgroup inserted.)

So the vertical asymptotes are .

2) Since , we need to find where . The cosine function is zero at for all integers ; thus the functions is undefined at .

## Horizontal Asymptotes

For rational functions in the form of where are both polynomials:

1. If the degree of is greater than that of the degree of , then the horizontal asymptote is at .

2. If the degree of is equal to that of the degree of , then the horizontal asymptote is at the quotient of the leading coefficient of over the leading coefficient of .

3. If the degree of is less than the degree of , see below (slanted asymptotes)

A function may not have more than one horizontal asymptote. Functions with a "middle section" may cross the horizontal asymptote at one point. To find this point, set y=horizontal asymptote and solve.

### Example Problem

Find the horizontal asymptote of .

#### Solution

The numerator has the same degree as the denominator, so the horizontal asymptote is the quotient of the leading coefficients:

## Slanted Asymptotes

Slanted asymptotes are similar to horizontal asymptotes in that they describe the end-behavior of a function. For rational functions , a slanted asymptote occurs when the degree of is one greater than the degree of . If the degree of is two or more greater than the degree of , then we get a curved asymptote. Again, like horizontal asymptotes, it is possible to get crossing points of slanted asymptotes, since again the slanted asymptotes just describe the behavior of the function as approaches .

For rational functions, we can find the slant asymptote simply by long division.

Hyperbolas have two slant asymptotes. Given a hyperbola in the form of , the equation of the asymptotes of the hyperbola are at (swap if the term is positive).