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 17:13, 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. Unknown error_msg)
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).
