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Unformatted text preview: 21 Rational Functions
Asymptotes and asymptotic behavior
Think about the function
1
y=
x
the various routes x can take, and the consequent behavior
of y:
as x goes
from 1 to +oo
from 1 to 0
from 1 to oo
from 1 to 0 sign of y
+
+
 magnitude of y
gets smaller and smaller
gets bigger and bigger
gets smaller and smaller
gets bigger and bigger In fact, the graph will look something like this: When a portion of a graph:
• extends infinitely in such a way as to get closer and
closer to a straight line without actually joining it
• we say that the line is an asymptote of the graph, and
• that the graph approaches the line asymptotically
21 p. 1 A rational function is a function of the form:
P (x )
ax + ...
f(x) =
=
Q( x )
bx +...
where P and Q are polynomials.
n m The key to understanding a rational function
lies in finding its asymptotes
VERTICAL ASYMPTOTES:
•
occur at zeros of denominator
•
graph cannot cross a vertical asymptote
•
it will go (offscale) to either +oo or  oo as it approaches a
vertical asymptote
HORIZONTAL ASYMPTOTE:
Three cases: Horizontal asymptote deg P > deg Q
deg P = deg Q
deg P < deg Q none
horizontal line y = a/b
xaxis •
graph will approach the horizontal asymptote both at
extreme left and extreme right of coordinate system 21 p. 2 Graphing a rational function
3n − 6
f ( n) =
Example:
n≥1
n −1
1. Draw in vertical asymptotes (at zeros of denominator).
zero of denominator n = 1 2. Draw in horizontal asymptote (if any).
degree of numerator = degree of denominator, so
y = 3 is horizontal asymptote. 3. Plot a few representative points
f(2) = 0, f(3) = 3/2, f(6) = 12/5, f(9) = 21/8 4. Draw.
• graph must approach horizontal asymptote at extremities (in this case extreme right).
•
graph must go offscale at vertical asymptotes
f(n)
4
3
2
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 n 20 −1
−2
−3
−4
−5 21 p. 3 ...
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This note was uploaded on 12/11/2011 for the course MATH 1324 taught by Professor Staff during the Spring '11 term at Austin Community College.
 Spring '11
 Staff
 Asymptotes, Rational Functions

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