“Onions have layers.”
3.4a Graphs
of Rational
Functions
Analyzing Rational Graphs
1) Factor if possible. Find the domain of the rational function.
2) Write the function in lowest terms (things cancel?)
3) Locate the x- and y-intercepts
4) Test graph for symmetry (optional)
5) Locate vertical asymptotes and holes
6) Locate horizontal or oblique asymptotes
7) Make a table of test points to help with graphing
8) Graph the rational function by hand
(use the graphing calculator to check your work!)
Analyzing Rational Graphs
2x 2  4 x  6
Analyze the graph of: f (x) 
2
x 9
1) Factor/Domain:
10
2) Lowest terms:

8
6
4
2
3) Intercepts:
-10 -8 -6 -4 -2
2
-2
4) Symmetry:
-4
-6
-8
-10
4
6
8 10
Analyzing Rational Graphs
2x  4 x  6 2(x  1)

Analyze the graph of: f (x) 
2
x 9
(x  3)
2
x3
5) VA and Holes:
10
6) HA or OA:
8

6
4
7) Test points:
2
-10 -8 -6 -4 -2
2
-2
8) Graph!
x-int: (-1,0) y-int: (0,2/3)
No symmetry
D: (-∞,-3)U(-3,3)U(3, ∞)
-4
-6
-8
-10
4
6
8 10
Analyzing Rational Graphs
Analyze the graph of:
3
f (x)  2
x 4
1) Factor/Domain:
10
2) Lowest terms:

8
6
4
3) Intercepts:
2
-10 -8 -6 -4 -2
4) Symmetry:
2
-2
-4
-6
-8
-10
4
6
8 10
Analyzing Rational Graphs
3
3

Analyze the graph of: f (x)  2
x  4 (x  2)(x  2)
5) VA and Holes:
10
6) HA or OA:

8
6
7) Test points:
4
2
-10 -8 -6 -4 -2
2
-2
8) Graph!
x-int: None y-int: (0,-3/4)
y-Axis Symmetry
D: (-∞,-2)U(-2,2)U(2, ∞)
-4
-6
-8
-10
4
6
8 10
Analyzing Rational Graphs
x 2  3x  2
Analyze the graph of: f (x) 
x 1
1) Factor/Domain:

2) Lowest terms:
3) Intercepts:
4) Symmetry:
Analyzing Rational Graphs
x 2  3x  2 (x  2)(x  1)
Analyze the graph of: f (x) 

x 1
x 1
5) VA and Holes:
6) HA or OA:

7) Test points:
8) Graph!
x-int: (-2,0) (-1,0) y-int: (0,-2)
No Symmetry
D: (-∞,1)U(1, ∞)
Analyzing Rational Graphs
x 3  2x 2  8x
Analyze the graph of: f (x)  2
x  5x  4
1) Factor/Domain:
2) Lowest terms:
3) Intercepts:
4) Symmetry:
5) VA and Holes:
6) HA or OA:
7) Test points:
8) Graph!

3.4a Graphs
of Rational
Functions
HW #6: p.208 #7 – 11 odd, 25, 29
Draw all graphs