Chabot Mathematics
§6.1 Rational Fcn
Mult & Div
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Review § 5.7
MTH 55
 Any QUESTIONS About
• §5.7 → PolyNomial Equation
Applications
 Any QUESTIONS About HomeWork
• §5.7 → HW-17
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Recall Rational Function
 A rational function is
a function, f(x), that is
a quotient of two
polynomials; i.e. 
p( x)
f ( x) 
,
q( x)
 Where
• where p(x) and q(x) are polynomials and
where q(x) is NOT the ZERO polynomial.
• The domain of f consists of all inputs x for
which q(x) ≠ 0.
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Rational FUNCTION Example
 RATIONAL FUNCTION ≡ a function
expressed in terms of rational expressions
 Example  Find f(3)
for this Rational
Function:
x  3x  7
f ( x) 
,
2
x 4
2
Chabot College Mathematics
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 SOLUTION
x 2  3x  7
f ( x) 
x2  4
32  3(3)  7
f (3) 
(3)2  4
997

95
11

4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Find the Domain of a Rational Fcn
1. Write an equation that sets the
denominator of the rational
function equal to 0.
2. Solve the denominator equation.
3. Exclude the value(s) found in step
2 from the function’s domain.
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Find Domain
 Find the Domain for
 SOLUTION
y3  5 y 2  4 y  0
y  y2  5 y  4  0
6
f ( y)  3
.
2
y  5y  4y
Set the denominator equal to 0.
Factor out the monomial GCF, y.
y  y  4 y  1  0 FOIL Factor the 2nd Degree polynomial
y  0 or y  4  0 or y  1  0 Use the zero products theorem.
y  4
y  1
 The fcn is undefined for y = 0, −4, or −1,
so the domain is {y|y  −4, −1, 0}.
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Find Domain
x
b. g x   2
 Find the Domain for
x  6x  8
 SOLUTION
• Find the values of x for
which the denominator x  2 x  4   0
x2 – 6x + 8 = 0, then
x  2  0 or x  4  0
exclude those values
x  2 or x  4
from the domain.
 The fcn is undefined for x = 2, or 4, so
the domain is {x|x  2, 4}.
• Interval Notation: (−∞,2)U(2,4)U(4,∞)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Graph
3
f ( x) 
x 1
 SOLUTION: x  1, so the graph has a
vertical asymptote at x = 1. Find
ordered pairs around the asymptote
and then graph.
x
y
4 2
3/5 1
0
3
0.5
6
x
y
1.5
6
4
1
5
3/4
Chabot College Mathematics
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2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Simplifying Rational
Expressions and Functions
 As in arithmetic, rational expressions
are simplified by “removing”, or
“Dividing Out”, a factor equal to 1.
 example
(2 x  1)( x  5) (2 x  1) ( x  5)


( x  7)( x  5)
( x  7) ( x  5)
(2 x  1)

( x  7)
Chabot College Mathematics
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equals 1
removed the factor
that equals 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Maintain Domain
 Because rational expressions often
appear when we are writing functions, it
is important that the function’s domain
not be changed as a result of
simplifying. For example,
(2 x  1)( x  5)
the Domain of the
F ( x) 
function given by
( x  7)( x  5)
is assumed to be all real numbers for
which the denominator is NONzero
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Maintain Domain
(2 x  1)( x  5)
 Thus for Rational Fcn: F ( x) 
( x  7)( x  5)
Domain of F  {x x  7, x  5}.
 In the previous example, we
(2 x  1)
.
wrote F(x) in simplified form as ( x  7)
 There is a serious problem with stating
that these are equivalent; The Domains
are NOT the same
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Maintain Domain
(2 x  1)( x  5)
 Why F ( x) 
( x  7)( x  5)
≠
(2 x  1)
.
( x  7)
 The domain of the function given by
(2 x  1)
G ( x) 
is assumed to be { x x  7}.
( x  7)
 Thus the domain of G includes 5, but
the domain of F does not. This problem
can be addressed by specifying
(2 x  1)( x  5) (2 x  1)

with x  5.
( x  7)( x  5) ( x  7)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Maintain Domain
 Write this Fcn in
Simplified form
3x 2  13x  10
g ( x)  2
3x  19 x  14
 SOLUTION: first factor the numerator
and denominator, looking for the largest
factor common to both. Once the
greatest common factor is found, use it
to write 1 and simplify as shown on the
next slide
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Maintain Domain
3x  13x  10
g ( x)  2
3x  19 x  14
Note that the domain of
g = {x | x  2/3 and x  −7}
by Factoring (see next)
(3x  2)( x  5)

( x  7)(3x  2)
Factoring. The greatest
common factor is (3x − 2).
(3x  2) ( x  5)


(3x  2) ( x  7)
Rewriting as a product of
two rational expressions.
( x  5)
 1
( x  7)
For x  2/3, we have
(3x − 2)/(3x − 2) = 1.
2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Maintain Domain
( x  5)

, x  23 .
( x  7)
Removing the factor 1. To
keep the same domain, we
specify that x  2/3.
2
3
x

13
x

10
 Thus the simplified g ( x)  2
3x  19 x  14
form of
x5
2
g ( x) 
, with x  3 .
x7
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
“Canceling” Confusion
 The operation of Canceling is a
ShortHand for DIVISION between
Multiplication Chains
 Canceling can ONLY be done when we
have PURE MULTIPLICATION
CHAINS both x 2  4 x  21 x 2  4 x  21 x  4  3


x7
x7
11
ABOVE &
BELOW the
x 2  4 x  21 x  3x  7 

 x  3
Division Bar`
x  7 
x7
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Canceling Caveat
 “Canceling” is a shortcut often used for removing a
factor equal to 1 when working with fractions.
Canceling removes multiplying factors equal to 1
in products. It cannot be done in sums or when
adding expressions together. Simplifying the
expression from the previous example might have
been done faster as follows:
3x 2  13x  10 (3x  2)( x  5) When a factor that

2
3x  19 x  14 ( x  7)(3x  2) equals 1 is found, it is
“canceled” as shown.
( x  5)

( x  7)
Chabot College Mathematics
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Removing a factor
equal to 1.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Canceling Caveat
Caution! Canceling is often performed incorrectly:
x7
5x  2 x  2
3
1
 7,

,

x
15
3
3  x x
Incorrect!
Incorrect!
Incorrect!
To check that
these are not
equivalent,
substitute a
number for x.
In each situation, the expressions canceled are
not both factors. Factors are parts of products.
For example, 5 is not a factor of the numerator
5x – 2. If you can’t factor, you can’t cancel!
When in doubt, do NOT cancel!
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Simplifying Rational Expressions
1. Write the numerator and
denominator in factored form.
2. Divide out all the common factors
in the numerator and denominator;
i.e., remove factors equal to ONE
3. Multiply the remaining factors in
the numerator and the remaining
factors in the denominator.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Simplify
 SOLUTION:
3  x 2  3x  4 
3x  9 x  12

2
6 x  30 x  24 6  x 2  5 x  4 
2
3  x  4  x  1

6  x  4  x  1
3x  4x  1

2  3x  4x  1
x4

2  x  4
Chabot College Mathematics
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3x 2  9 x  12
6 x 2  30 x  24
Factor out the GCF.
Factor the
polynomial factors.
Divide out
common factors.
and x  1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Multiply Rational Expressions
 The Product of Two Rational
Expressions
 To multiply rational expressions,
multiply numerators and multiply
denominators:
P R PR
 
Q S QS
 Then factor and simplify the
result if possible.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Multiplication
 Multiply and, if possible, simplify.
x2  5x  4 x  3

b)
2
x 9
x4
6a 4 5

a)
10 6a
 SOLUTION a)
6a 4 5 6a 4 (5)


10 6a 10(6a)
2 3 a  a  a  a 5

2 5 2 3 a
MULTIPLICATION
Chains → Canceling OK
2  3  a  a  a  a  5 a3


2
2  5  2 3  a
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Multiplication
x2  5x  4 x  3
 SOLUTION b)

2
x 9
x4
x 2  5x  4 x  3 ( x 2  5 x  4)( x  3)


2
2
x 9
x4
( x  9)( x  4)
( x  4)( x  1)( x  3)

( x  3)( x  3)( x  4)

Chabot College Mathematics
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MULTIPLICATION
Chains →
Canceling OK
( x  4) ( x  1) ( x  3)
( x  3) ( x  3) ( x  4)
x 1

x3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Multiply & Simplify
 Multiply and,
if possible, simplify.
 SOLUTION
x 2  x  20
7
 2
21
3x  11x  4


x 2  x  20
7
x 2  x  20 7
 2

21
3x  11x  4 21 3x 2  11x  4
( x  4)( x  5)7

7(3)(3x  1)( x  4)

Chabot College Mathematics
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

MULTIPLICATION
Chains →
Canceling OK
( x  4) ( x  5) 7
7 (3)(3 x  1) ( x  4)
( x  5)

3(3x  1)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Divide Rational Expressions
 The Quotient of Two
Rational Expressions
 To divide by a rational expression,
multiply by its reciprocal
V Y V Z VZ
   
W Z W Y WY
 Then factor and, if possible, simplify.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Division
 Divide and, if possible, simplify.
a)
x 8

9 y
 SOLUTION
x 8 x y
a)   
9
b)
y
Multiplying by the reciprocal of the divisor
9 8
xy

Multiplying rational expressions
72
x2 x3 x5
x 3


x5
1 x2
( x  3)( x  5)

x2
Chabot College Mathematics
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b)
x2
x 3
x5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Division
 Divide and,
if possible, simplify.
 SOLN
x2
x2
 2
2
x  4 x  4x  4
x2
x2
x  2 x2  4x  4
 2
 2

2
x  4 x  4x  4 x  4
x2
MULTIPLICATION
Chains → Canceling OK
( x  2)( x  2)( x  2)

( x  2)( x  2)( x  2)

Chabot College Mathematics
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( x  2) ( x  2) ( x  2)
( x  2) ( x  2) ( x  2)
x2

x2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Division
 Divide and,
if possible, simplify.
x 2  3x  2
x2  5x  6
 2
2
x  5 x  4 x  10 x  24
x 2  3x  2
x2  5x  6
x 2  3x  2 x 2  10 x  24
 2
 2
 2
2
x  5 x  4 x  10 x  24 x  5 x  4 x  5 x  6

 SOLUTION
MULTIPLICATION
Chains → Canceling OK
Chabot College Mathematics
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( x  1)( x  2)( x  6)( x  4)

( x  4)( x  1)( x  2)( x  3)

( x  1) ( x  2) ( x  6) ( x  4)
( x  4) ( x  1) ( x  2) ( x  3)
( x  6)

( x  3)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Division
 Divide and,
if possible, simplify.
x 2  3x  4 x  1

2
x  25
x6
x 2  3x  4 x  1 x 2  3x  4 x  6



2
2
x  25
x6
x  25 x  1

 SOLUTION
( x  4)( x  1)( x  6)

( x  5)( x  5)( x  1)

MULTIPLICATION
Chains →
Canceling OK
( x  4) ( x  1) ( x  6)
( x  5)( x  5) ( x  1)
( x  4)( x  6)

( x  5)( x  5)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Manufacturing Engr
 The function
given by
3t 2  2t
M (t ) 
4t  2
 gives the time, in hours, for two machines,
working together to complete a job that the
1st machine could do alone in t hours and
the 2nd machine could do in 3t − 2 hours.
• How long will the two machines, working
together, require for the job if the first machine
alone would take (a) 2 hours? (b) 5 hours?
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Example  Manufacturing Engr
 SOLUTION
(a) M (2) 
3 2  2  2
42  2
2
12  4 8 4

 
or 1 13 hr
82 6 3
3  52  2  5
(b) M (5) 
45  2
Chabot College Mathematics
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75  10 65
11


or 3 18
hr
20  2 18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
WhiteBoard Work
 Problems From §6.1 Exercise Set
• 22 (ppt), 26 (ppt), 114 , 16, 46, 66, 86

More Rational Division
 Since we are dividing fractions,
we multiply by the reciprocal
 Now, we follow the rule for
multiplication
 Factor and then cancel
 Don't leave the numerator empty
- put a one to hold the place.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
P6.1-22  Rational fcn Graph
 Describe
end-behavior of
Graph at Far-Right
 ANS: As x becomes
large y = f(x)
approaches, but
never reaches,
a value of 3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
P6.1-22  Rational fcn Graph
 What is the Eqn for
the Horizontal
Asymptote:
 ANS: y = f(x)
approaches, but
never reaches, a
value of 3, to the
Asymptote eqn
y=3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
P6.1-26  Rational fcn Graph
 List 2 real No.s that
are NOT function
values of f
 ANS: y = f(x) does
not have a graph
between y > 0 and
y < 3. Thus two
values for which
there is NO f(x):
• y = 1 or y = 2
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
P6.1-114  Smoking Diseases
 Find P(9). Describe
Meaning. ID pt on
Graph
1009  1 800
P9 

 88.9
9
9
 ANS: An incidence
ratio of 9 indicates
that 88.9% of
Lung Cancer deaths are associated
with Cigarette smoking
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
P6.1-114  Smoking Diseases
 ID P(9)
on Graph
88.9
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
All Done for Today
 The FIRST Algebraist
Diophantus
of
Alexandria
Chabot College Mathematics
38
• In the 3rd century, the
Greek mathematician
Diophantus of
Alexandria wrote his
book Arithmetica. Of the
13 parts originally
written, only six still
survive, but they provide
the earliest record of an
attempt to use symbols
to represent unknown
quantities.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
40
5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
4
5
6
y
5
2
y
1
x
4
0
-6
-5
-4
-3
-2
-1
3
0
1
2
3
4
-1
-2
2
-3
1
-4
x
-5
5
-6
0
-3
-2
-1
0
1
2
3
-1
4
-7
-8
-2
-9
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
41
file =XY_Plot_0211.xls
-10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt
5
6