2-1 Part II: Graphing Rational Functions
Six steps to pretty good graph of a rational function
x2
Example:
f ( x)  2
x x
1. Factor numerator and denominator as much as possible.
x2
f ( x) 
x ( x  1)
2. Draw in vertical asymptotes (at zeros of denominator).
zeros of denominator are 0 and 1.
3. Draw in horizontal asymptote (if any).
degree of numerator < degree of denominator, so
x-axis is horizontal asymptote.
vertical asymptotes
f(x)
10
horizontal asymptote
x
-3
2
2-1 Part II
p. 1
4. Draw in zeros of function (zeros of numerator). x = 2
5. Add guide points between and beyond each zero and vertical
asymptote. f(-2) = -2/3
f(1/2) = 6 f(3/2) = -2/3 f(3) = 1/6
guide points
x
X
X
-3
X
X
X
2
zero
Note: there is one zero and two vertical asymptotes for a
total of three. You will need 3 + 1 = 4 guide points. You
always need one more guide point than the total of zeros
and vertical asymptotes. And that should be enough to
ensure a good-enough graph.
2-1 Part II
p. 2
6. Draw. Make sure function approaches horizontal
asymptote at left and right extremities.




start at extreme left
graph must enter from close to the horizontal asymptote there
or come from + or - oo, if no horizontal asymptote
will the graph start from above the asymptote or below the
asymptote?
 look at the left-most guide point for the clue
 you must start on that side of the asymptote
 now draw the graph through the left-most "in-between" point
 graph must go off-scale at vertical asymptotes
 it must never cross the x-axis except at marked zeros
This can all be done with almost no algebra; just draw
in some asymptotes and points and draw your graph!
f(x)
-3
2-1 Part II
10
2
p. 3