1.
2.
3.
4.
Objectives:
Be able to identify the parent function for a rational.
Be able list the characteristic of a rational function.
Be able to graph rational functions in general form.
Be able to graph rational functions in polynomial form.
Critical Vocabulary:
Parent function, Rational Function, Asymptote
I. The Parent Function
Parent Functions: f ( x ) 
1
x
This function will have a
vertical asymptote at x = 0.
This function will have a
horizontal asymptote at y = 0.
This function will be a
hyperbola (which consists of 2
symmetrical parts called
branches.
Domain: All Real #; except x ≠ 0
Range: All Real #; except y ≠ 0
II. The Rational Function
a
k
xh
a: Determines the size and direction
Parent Functions: f ( x) 
If a is positive the graph will be in
sections 1 and 3.
If a is negative the graph will be in
sections 2 and 4.
lal > 1: Hyperbolas change slower
lal < 1: Hyperbolas change quicker
h: horizontal shift
(Vertical Asymptote: x = #)
K: Vertical Shift
(Horizontal Asymptote: y = #)
f ( x) 
17111
3
(4xxx 3)
III. Graphing a Rational Function (General Form)
3
3
Example 1: Graph f ( x) 
( x  2)
1st: List the Characteristics:
•Hyperbolas in S1 and S3
•Slow Change
•VA: x = -2
•HA: y = -3
2nd: Graph your asymptotes
3rd: Find Two more points
x
1
-5
y
-2
-4
4th: Find the Domain and Range
D: All Real #; except x ≠ -2
R: All Real #; except y ≠ -3
III. Graphing a Rational Function (General Form)
4
3
Example 2: Graph f ( x) 
( x  1)
1st: List the Characteristics:
•Hyperbolas in S2 and S4
•Slow Change
•VA: x = 1
•HA: y = 3
2nd: Graph your asymptotes
3rd: Find Two more points
x
5
-3
y
2
4
4th: Find the Domain and Range
D: All Real #; except x ≠ 1
R: All Real #; except y ≠ 3
Page 561 #12, 13, 19, 21
a. List the Characteristics
b. Graph (show table)
c. Find Domain and Range
1.
2.
3.
4.
Objectives:
Be able to identify the parent function for a rational.
Be able list the characteristic of a rational function.
Be able to graph rational functions in general form.
Be able to graph rational functions in polynomial form.
Critical Vocabulary:
Parent function, Rational Function, Asymptote
Warm Up: Graph the following:
1. f ( x) 
2
4
( x  3)
IV. Graphing a Rational Function (Polynomial Form)
2x 1
f ( x) 
Example 1: Graph
x3
1st: Find (and graph) your asymptotes
VA: Place where the function is und.
x-3=0
x=3
HA: Leading co-efficient of numerator
divided by the leading co-efficient
of the denominator.
y=2
3rd: Find Two more points
x
4
2
y
9
-5
4th: Find the Domain and Range
D: All Real #; except x ≠ 3
R: All Real #; except y ≠ 2
IV. Graphing a Rational Function (Polynomial Form)
3x  6
f
(
x
)

Example 2: Graph
x2
1st: Find (and graph) your asymptotes
VA: Place where the function is und.
x+2=0
x = -2
HA: Leading co-efficient of numerator
divided by the leading co-efficient
of the denominator.
y=3
3rd: Find Two more points
x
0
-4
y
-3
9
4th: Find the Domain and Range
D: All Real #; except x ≠ -2
R: All Real #; except y ≠ 3
Page 562 #27, 28, 30
(3 problems)