Objectives:1. Be able to identify the parent function for a rational.2. Be able list the characteristic of a rational function.3. Be able to graph rational functions in general form.4. Be able to graph rational functions in polynomial form.

Critical Vocabulary:Parent function, Rational Function, Asymptote

I. The Parent Function

xxf

1)( Parent Functions:

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

khx

axf

)(Parent Functions:

a: Determines the size and direction

h: horizontal shift (Vertical Asymptote: x = #)K: Vertical Shift (Horizontal Asymptote: y = #)

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.

xxf

1)(

xxf

1)(

lal > 1: Hyperbolas change slower

xxf

7)(

lal < 1: Hyperbolas change quicker

xxf

4

1)(

)3(

1)(

x

xf 31

)( x

xf

III. Graphing a Rational Function (General Form)

3)2(

3)(

x

xfExample 1: Graph

1st: List the Characteristics:

•Hyperbolas in S1 and S3

•Slow Change•VA: x = -2•HA: y = -32nd: Graph your

asymptotes3rd: Find Two more points

x

y

1

-2

-5

-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)

3)1(

4)(

x

xfExample 2: Graph

1st: List the Characteristics:

•Hyperbolas in S2 and S4

•Slow Change•VA: x = 1•HA: y = 32nd: Graph your

asymptotes3rd: Find Two more points

x

y

5

2

-3

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 Characteristicsb. Graph (show table)c. Find Domain and Range

Objectives:1. Be able to identify the parent function for a rational.2. Be able list the characteristic of a rational function.3. Be able to graph rational functions in general form.4. Be able to graph rational functions in polynomial form.

Critical Vocabulary:Parent function, Rational Function, Asymptote

Warm Up: Graph the following:

4)3(

2)(.1

x

xf

IV. Graphing a Rational Function (Polynomial Form)

3

12)(

x

xxfExample 1: Graph

1st: Find (and graph) your asymptotes

3rd: Find Two more points

x

y

4

9

2

-5

4th: Find the Domain and Range

D: All Real #; except x ≠ 3 R: All Real #; except y ≠ 2

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

IV. Graphing a Rational Function (Polynomial Form)

2

63)(

x

xxfExample 2: Graph

1st: Find (and graph) your asymptotes

3rd: Find Two more points

x

y

0

-3

-4

9

4th: Find the Domain and Range

D: All Real #; except x ≠ -2 R: All Real #; except y ≠ 3

VA: Place where the function is und.x + 2 = 0

x = -2HA: Leading co-efficient of numerator divided by the leading co-efficient of the denominator.

y = 3

Page 562 #27, 28, 30(3 problems)