Asymptotes and asymptotic behavior
Think about the function y =
1 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
2-1 p. 1

A rational function is a function of the form:
 f(x) = = ax n ...
bx m

...
where P and Q are polynomials.
The key to understanding a rational function
lies in finding its asymptotes
V ERTICAL ASYMPTOTES :
 occur at zeros of denominator
 graph cannot cross a vertical asymptote
 it will go (off-scale) to either + oo or - oo as it approaches a vertical asymptote
H ORIZONTAL ASYMPTOTE :
Three cases: Horizontal asymptote deg P > deg Q deg P = deg Q none horizontal line y = a/b deg P < deg Q x-axis
 graph will approach the horizontal asymptote both at extreme left and extreme right of coordinate system
2-1 p. 2

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2-1
Graphing a rational function
Example: f ( n )

3 n

6
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 off-scale at vertical asymptotes f(n)
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