Chabot Mathematics
§5.1 Intro to
PolyNomials
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Review § 4.3
MTH 55
 Any QUESTIONS About
• §4.3 → Absolute Value: Equations &
InEqualities
 Any QUESTIONS About HomeWork
• §4.3 → HW-14
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Mathematical “TERMS”
 A TERM can be a number, a variable, a
product of numbers and/or variables, or
a quotient of numbers and/or variables.
 A term that is a product of constants
and/or variables is called a monomial.
• Examples of monomials: 8,
w,
24x3y
 A polynomial is a monomial or a sum
of monomials. Examples of polynomials:
• 5w + 8, −3x2 + x + 4,
Chabot College Mathematics
3
x,
0,
75y6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Terms
 Identify the terms of the polynomial
7p5 − 3p3 + 3
 SOLUTION
 The terms are 7p5, −3p3, and 3.
• We can see this by rewriting all
subtractions as additions of opposites:
7p5 − 3p3 + 3 = 7p5 + (−3p3) + 3
These are the terms of the polynomial.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
[Bi, Tri, Poly]-nomials
 A polynomial that is composed of two
terms is called a binomial, whereas
those composed of three terms are
called trinomials. Polynomials with four
or more terms have no special name
Monomials Binomials Trinomials
Polynomials
5x2
3x + 4
3x2 + 5x + 9
5x3  6x2 + 2xy  9
8
4a5 + 7bc 7x7  9z3 + 5
a4 + 2a3  a2 + 7a  2
8a23b3
10x3  7 6x2  4x  ½ 6x6  4x5 + 2x4  x3 + 3x  2
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Polynomial DEGREE
 The degree of a term of a polynomial is
the no. of variable factors in that term
 EXAMPLE: Determine the degree of
each term: a) 9x5
b) 6y
c) 9
 SOLUTION
 a) The degree of 9x5 is 5
 b) The degree of 6y (6y1) is 1
 c) The degree of 9 (9z0) is 0
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Mathematical COEFFICIENT
 The part of a term that is a constant
factor is the coefficient of that term.
The coefficient of 4y is 4.
 EXAMPLE: Identify the coefficient of each
term in polynomial: 5x4 − 8x2y + y − 9
 SOLUTION
 The coefficient of 5x4 is 5.
 The coefficient of −8x2y is −8.
 The coefficient of y is 1, since y = 1y.
 The coefficient of −9 is simply −9
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
DEGREE of POLYNOMIAL
 The leading term of a polynomial is the
term of highest degree. Its coefficient is
called the leading coefficient and its
degree is referred to as the degree of
the polynomial.
 Consider this polynomial
4x2 – 9x3 + 6x4 + 8x – 7.
• Find the TERMS, COEFFICIENTS,
and DEGREE
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
DEGREE of POLYNOMIAL
 For polynomial: 4x2 − 9x3 + 6x4 + 8x − 7
• List Terms, CoEfficients, Term-Degree
Terms → 4x2, −9x3, 6x4, 8x, and −7
coefficients → 4, −9, 6, 8 and −7
degree of each term → 2, 3, 4, 1, and 0
The leading term is 6x4 and the
leading coefficient is 6.
 The degree of the polynomial is 4.




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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  −3x4 + 6x3 − 2x2 + 8x + 7
 Complete Table for PolyNomial
–3x4 + 6x3 – 2x2 + 8x + 7
Term PolyNomial
Term Coefficient
Degree
Degree
–3
6x3
2
1
7
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  –3x4 + 6x3 – 2x2 + 8x + 7
 Put Terms in
Descending Exponent Order
Term PolyNomial
Term Coefficient
Degree
Degree
–3x4
–3
6x3
–2x2
2
8x
1
7
7
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  –3x4 + 6x3 – 2x2 + 8x + 7
 Coefficients are the CONSTANTS
before the Variables
Term PolyNomial
Term Coefficient
Degree
Degree
–3x4
−3
6x3
6
–2x2
–2
2
8x
8
1
7
7
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  –3x4 + 6x3 – 2x2 + 8x + 7
 Term DEGREE is the
Value of the EXPONENT
Term PolyNomial
Term Coefficient
Degree
Degree
–3x4
–3
4
6x3
6
3
–2x2
–2
2
8x
8
1
7
7
0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  –3x4 + 6x3 – 2x2 + 8x + 7
 Polymomial Degree is the SAME as
the highest Term Degree
Term PolyNomial
Term Coefficient
Degree
Degree
–3x4
–3
4
6x3
6
3
–2x2
–2
2
8x
8
1
7
7
0
4
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
MultiVariable PolyNomials
 Evaluate the 2-Var polynomial
5 + 4x + xy2 + 9x3y2 for x = −3 & y = 4
 Solution: Substitute −3 for x and 4 for y:
5 + 4x + xy2 + 9x3y2
= 5 + 4(−3) + (−3)(4)2 + 9(−3)3(4)2
= 5 − 12 − 48 − 3888
= −3943
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Degree of MultiVar Polynomial
 Recall that the degree of a polynomial is the
number of variable factors in the term.
 Example: ID the coefficient and the degree of each
term and the degree of the polynomial
10x3y2 – 15xy3z4 + yz + 5y + 3x2 + 9
Term
10x3y2
10
5
–15xy3z4
–15
8
yz
1
2
5y
5
1
3x2
3
2
9
9
0
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Coefficient Degree
Degree of the
Polynomial
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Like Terms
 Like, or similar terms either have exactly
the same variables with exactly the
same exponents or are constants.
 For example,
9w5y4 and 15w5y4 are like terms
 and
−12 and 14 are like terms,
 but
−6x2y and 9xy3 are not like terms.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Combine Like Terms
10x2y + 4xy3 − 6x2y − 2xy3
8st − 6st2 + 4st2 + 7s3 + 10st − 12s3 + t − 2
SOLUTION
10x2y + 4xy3 − 6x2y − 2xy3
= (10 − 6)x2y + (4 − 2)xy3
= 4x2y + 2xy3
b) 8st − 6st2 + 4st2 + 7s3 + 10st − 12s3+ t − 2
= −5s3 − 2st2 + 18st + t − 2
a)
b)

a)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Common Properties: PolyNom Fcns
1. The domain of a polynomial
function is the set of
all real numbers.
2. The graph of a polynomial function
is a continuous curve.
• This means that the graph has no
holes or gaps and can be drawn on a
piece of paper without lifting the pencil.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Continuous vs. DisContinuous
Could be a
PolyNomial
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Can NOT be
a PolyNomial
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Common Properties: PolyNom Fcns
3. The graph of a polynomial function
is a smooth curve.
• This means that the graph of a
polynomial function does NOT contain
any SHARP corners.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Smooth vs. Kinked/Cornered
Could be a
PolyNomial
Chabot College Mathematics
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Can NOT be
a PolyNomial
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Leading Coefficient Test
 Given a PolyNomial Function of the form
n
n1
f x   an x  an1 x  ...  a1 x  a0 a  0
 The leading term is anxn. The behavior of
the graph of f(x) as x →  or as
x → − is dominated by this term, and
is similar to one of the following 4 graphs
• Note that The middle portion of each graph,
indicated by the dashed lines,
is NOT determined by this test.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Lead Coeff Test: Odd & Positive
1. Leading Term
•
ODD Exponent
•
POSITIVE
Coeff
e.g. : f x   7 x9  13x 4  96
 Graph
• FALLS to LEFT
• RISES to RIGHT
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Lead Coeff Test: Odd & Negative
2. Leading Term
•
ODD Exponent
•
NEGATIVE
Coeff
e.g. : f x   8x9  12 x 4  77
 Graph
• RISES to LEFT
• FALLS to RIGHT
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Lead Coeff Test: Even & Positive
3. Leading Term
•
EVEN Exponent
•
POSITIVE
Coeff
e.g. : f x   23x8  5x5  88
 Graph
• RISES to LEFT
• RISES to RIGHT
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Lead Coeff Test: Even & Negative
4. Leading Term
•
EVEN Exponent
•
NEGATIVE
Coeff
 Graph
• FALLS to LEFT
• FALLS to RIGHT
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e.g. : f x   32 x8  7 x5  9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Lead CoEff Test
 Use the leading-CoEfficient test to
determine the end behavior of the graph of
3
2
y  f x   2x  3x  4.
 SOLUTION
• Here n = 3 (odd) and an = −2 < 0. Thus,
Case-2 (Odd & Neg) applies. The graph of
f(x) rises to the left and falls to the right.
This behavior is described by: y → 
as x → −; and y → − as x → 
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Adding Polynomials
 EXAMPLE  Add
(−6x3 + 7x − 2) + (5x3 + 4x2 + 3)
 Solution → Combine Like terms
(−6x3 + 7x − 2) + (5x3 + 4x2 + 3)
= (−6 + 5)x3 + 4x2 + 7x + (−2 + 3)
= −x3 + 4x2 + 7x + 1
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Add Polynomials
 Add:
(3 – 4x + 2x2) + (–6 + 8x – 4x2 + 2x3)
 Solution
(3 – 4x + 2x2) + (–6 + 8x – 4x2 + 2x3)
= (3 – 6) + (–4 + 8)x + (2 – 4)x2 + 2x3
= –3 + 4x – 2x2 + 2x3
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Add Polynomials
 Add: 10x5 – 3x3 + 7x2 + 4 and
6x4 – 8x2 + 7 and 4x6 – 6x5 + 2x2 + 6
 Solution
10x5
- 3x3 + 7x2 + 4
6x4
- 8x2 + 7
4x6 - 6x5
+ 2x2 + 6
4x6 + 4x5 + 6x4 - 3x3 + x2 + 17
 Answer: 4x6 + 4x5 + 6x4 − 3x3 + x2 + 17
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Opposite of a PolyNomial
 To find an equivalent polynomial
for the opposite, or additive
inverse, of a polynomial, change
the sign of every term.
• This is the same as multiplying the
original polynomial by −1.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Opposite of PolyNom
 Simplify:
–(–8x4 – x3 + 9x2 – 2x + 72)
 Solution
–(–8x4 – x3 + 9x2 – 2x + 72)
= (–1)(–8x4 – x3 + 9x2 – 2x + 72)
= 8x4 + x3 – 9x2 + 2x – 72
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
PolyNomial Subtraction
 We can now subtract one
polynomial from another by
adding the opposite of the
polynomial being subtracted.
PolyNomial Subtractor
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Subtract PolyNom
 (10x5 + 2x3 – 3x2 + 5) – (–3x5 + 2x4 – 5x3 – 4x2)
 Solution
(10x5 + 2x3 – 3x2 + 5) – (–3x5 + 2x4 – 5x3 – 4x2)
= 10x5 + 2x3 – 3x2 + 5 + 3x5 – 2x4 + 5x3 + 4x2
= 13x5 – 2x4 + 7x3 + x2 + 5
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Subtract
 (8x5 + 2x3 – 10x) – (4x5 – 5x3 + 6)
 Solution
(8x5 + 2x3 – 10x) – (4x5 – 5x3 + 6)
= 8x5 + 2x3 – 10x + (–4x5) + 5x3 – 6
= 4x5 + 7x3 – 10x – 6
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
Example  Column Form
 Write in columns and subtract:
(6x2 – 4x + 7) – (10x2 – 6x – 4)
 Solution
6x2 – 4x + 7
–(10x2 – 6x – 4)
–4x2 + 2x + 11
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
WhiteBoard Work
 Problems From §5.1 Exercise Set
• By ppt → 22, 24, 26, 28, 70
• 10
Adding and Subtracting Functions
If f(x) and g(x) define functions, then
and
(f + g) (x) = f (x) + g(x)
(f – g) (x) = f (x) – g(x).
Sum function
Difference function
In each case, the domain of the new function is the
intersection of the domains of f(x) and g(x).
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
P5.1-[22, 24]
 PolyNomial or
NOT PolyNomial
KINKED → NOT
a Polynomial
SMOOTH &
CONTINUOUS →
IS a Polynomial
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
P5.1-[26, 28]
 Use Lead
CoEfficient Test
of End Behavior
to Match Fcn to
Graph
Odd & Pos →
Falls-Lt & Rises-Rt
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Odd & Negs →
Rise-Lt & Falls-Rt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
P5.1-70  AIDS Mortality Models
 Given PolyNomial Models for USA
AIDS mortality over the years
1990-2002 where x ≡ yrs since 1990
f x   1844 x  54925x  111568
3
2
g x   11x  2066 x  56036 x  110590
2
 Bar Chart shows ACTUAL 2002
Mortality of 501 669
 Find Error Associated with Each Model
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
P5.1-70  AIDS Mortality Models
 Evaluate Model
using MATLAB
Math-Processing
Software
• See MTH25 for
Info on MATLAB
>> x =2002-1990
x = 12
>> fx = -1844*x^2 + 54923*x +
111568
fx = 505108
>> gx = -11*x^3 - 2066*x^2 +
56036*x + 110590
gx =
 By MATLAB the
Model Errors
• f(x) → 0.69% Low
• g(x) → 7.0% Low
Chabot College Mathematics
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466510
>> Yactual = 501669
>> fx_error = (fx-Yactual)/Yactual
fx_error = 0.0069 = 0.69%
>> gx_error = (gx-Yactual)/Yactual
gx_error =
-0.0701 = -7.01%
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
All Done for Today
Lead Coeff
Test
Summarized
Chabot College Mathematics
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n is Even
n is Even
an > 0
an < 0
n is Odd
n is Odd
an > 0
an < 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.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
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.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
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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-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt
4
5
6
y
5
4
3
2
1
x
0
-3
-2
-1
0
1
2
3
4
5
-1
-2
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-20_sec_5-1_Intro_to_PolyNom_Fcns.ppt