If , where P(x) and Q(x) are polynomial functions, then f(x) is said to be a rational funtion.
Objective: We want to know how to graph a rational function.
Example 1: Let Graph f.
(1) Vertical Asymptote(s).
First, we set denominators we get x=1 and x=2, these are called the vertical asymtotes of f. In short, these are vertical lines that the graph of f will get very close to. So let's investigate the following cases:
(a) As (this means x approaches to 1 from the right, say x=1.0001), we get (this measns that the outputs will tend to
(b) As (this means x approaches to 1 from the left, say x=0.999, note that it does not mean x approaches to -1), we get
(c) As say x=2.0001, we get
(d) As say x=1.9999, we get
(2) Horizontal Asymptote:
As , say x=105, and as say x=-105, Therefore the horizontal asymtote for f is y=0.
(3) Find the interval(s) where f is increasing or decreasing.
We rewrite (note. I mistyped "3", it should be "2") so ("3" should be "2" here again) (chain rule).
Thus, ("3" should be "2" again) Now we need to make a table for (because is not in standard form, so we can't apply the short cut to solve the inequality).
Thus, f is increasing in and decreasing in and f has a maximum at (note: it should be increasing in (-infinity, 3/2) union (2, infinity). It decreases in (3/2,2).)
(4) Sketch the graph of f.
Let's use to graph the function. Maple syntax: plot(2/((x-1)*(x-2)), x = -1..3, y = -50..50);