Graphing Simple and
General Rational
Functions
Section 9.2 and 9.3
WHAT YOU WILL LEARN:
1.
How to graph basic rational (fraction) functions.
2. How to graph general (or more complex) rational
functions.
Oh Boy! Vocabulary
Rational Function – is a function of the form:
f ( x) 
p( x)
q( x)
Where p(x) and q(x) are polynomials and q(x)
does not equal zero (why?).
The Most Basic (or Parent) Graph
The graph of
like:
y
1
x
is a hyperbola that looks
y
The x-axis is a horizontal
asymptote (why?)
10
The y-axis is a vertical
asymptote.
5
-10
-5
5
10
x
The domain and range are all
non-zero real numbers.
The two symmetrical parts are
called branches.
-5
-10
Get Your Calculator’s Ready
• Graph the following:
y 
1
x
y 
2
x
y 
3
x
y 
What do you notice?
1
x
Some More Rules/Information
All rational functions of the form:
y
a
xh
k
have graphs that are hyperbolas with asymptotes
at x = h and y = k.
To draw the graph, figure out where the
asymptotes are and then use an x/y table to get a
couple of points on either side of the vertical
asymptote.
Example
• Graph:
y
2
x3
1
y
10
5
-10
-5
5
-5
-10
10
x
You Try
• Graph
y
3
x 1
2
y
10
5
-10
-5
5
-5
-10
10
x
More Information!
All rational functions of the form
y
ax  b
cx  d
also have graphs that are hyper-bolas. The
vertical asymptote occurs at the x-value
that makes the denominator zero. The
horizontal asymptote is the line: y  a
c
Example
• Graph
y
x 1
state the domain and range.
2x  4
y
10
5
-10
-5
5
-5
-10
10
x
Let’s Try #34 on page 544
• Graph
y
x7
3x  8
y
10
5
-10
-5
5
-5
-10
10
x
9.3 Graphing General Rational Functions
We are now going to graph more “complex” rational functions.
They will be more of the type:
p( x)
somepolyno mial
f ( x) 

q( x)
anotherpol ynomial
The x-intercepts of the graph of f are the real zeros of p(x).
The graph of f has a vertical asymptote at each real zero of
q(x)
Degree of numerator less than degree of denominator, y = 0 is
horizontal asymptote.
Degree of numerator equal to degree of denominator,
horizontal asymptote is at coefficient of leading term of
numerator divided by coefficient of leading term of
denominator.
Degree of numerator is greater than degree of denominator,
then no horizontal asymptote.
Graphing Example
• Graph:
y
4
. State the domain and range.
x 1
2
• The numerator has no zeros, so there is no xintercept.
• The denominator has no real zeros so there is no
vertical asymptote.
• The degree of the numerator is less than the
degree of the denominator so the line y = 0 is a
horizontal asymptote.
Graphing Example m<n
• Graph:
y
4
. State the domain and range.
x 1
2
y
10
5
-10
-5
5
-5
-10
10
x
You Try
• Graph
y
x
x 1
2
y
10
5
-10
-5
5
-5
-10
10
x
Another Example m = n
• Graph:
y
3x
2
x 4
2
• The numerator has 0 as its only zero, so there is
one x-intercept at (0,0).
• The denominator can be factored to (x+2) (x-2) so
the denominator has two zeros at x=2 and x=-2
are vertical asymptotes.
• The degree of the numerator is the same as the
degree of the denominator so there is a
a 3
horizontal asymptote at
y 
b 1
Another Example m = n
• Graph:
y
3x
2
x 4
y
2
10
5
-10
-5
5
-5
-10
10
x
Let’s Do #28 Together
• Graph
x 4
2
y
x 3
y
2
10
5
-10
-5
5
-5
-10
10
x
A Side Note
• We aren’t going to worry about the case where the
degree of the numerator is greater than the degree
of the denominator…we’ll see that in Trig/Pre-Calc.
Homework
Homework: page 543, 12, 16, 20-22 all, 24, 32,
page 550, 26 and 30 (identify asymptotes)