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
[hide]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:
$
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).