Asymptote (geometry): Difference between revisions
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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).
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).
