algebra 2: section 6.2
TRANSCRIPT
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Algebra 2: Section 6.2
Evaluating and Graphing Polynomial Functions
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Polynomial Function
• A function is a polynomial function if…
– Exponents are all whole numbers
– Coefficients are all real numbers
• Standard Form of Polynomial Function
– All terms are written in descending order of
exponents from left to right
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1 1 0( ) n n
n nf x a x a x a x a
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Parts of Polynomial
Function
• Leading coefficient
– Coefficient on highest power of x
• Constant term
– Term that has no variable (no x)
• Degree of the polynomial
– Exponent of the highest power of x
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Classifying Polynomial
Functions
• Classify based on highest power of x
• Power of x
– Zero: Constant
– One: Linear
– Two: Quadratic
– Three: Cubic
– Four: Quartic
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Examples
2 21. ( ) 2f x x x
No, because negative exponent.
• Decide whether the function is a
polynomial function. If it is, write the
function in standard form and state its
degree, type, and leading coefficient.
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Examples
3 42. ( ) 0.8 5g x x x
Yes4 3( ) 0.8 5f x x x
Degree: 4
Type: Quartic
Leading Coefficient: 1
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Synthetic Substitution
(Synthetic Division)
• Gives another way to evaluate a function
• Also used to divide polynomials
– This will be discussed in later sections
• The last entry is the value of the function
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Examples
• Use synthetic division to evaluate.
5 43. ( ) 3 5 10 when 2f x x x x x
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5 43. ( ) 3 5 10 when 2f x x x x x
3 1 0 0 5 102
3
6
7
14
14
28
28
56
51
102
92
Coefficients of x written in order
Missing power of
x, zero coefficient!
Number you are
evaluating goes in
front
Drop 1st
number
down
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3 24. ( ) 5 4 1; (4)f x x x x f
5 1 4 14
5
20
21
84
80
320
321
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Assignment
• p.333
#15-26 all, 37-46 all
(22 problems)
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What happens to the graph when x gets
very small or x gets very large?
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As x gets very small
the graphs is falling
As x gets very large
the graphs is rising
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What happens to the graph when x gets
very small or x gets very large?
13
As x gets very small
the graphs is rising
As x gets very large
the graphs is rising
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End Behavior of Graphs of
Polynomial Functions
• What the function’s graph does as x approaches positive
and negative infinity
x
( )f x
x“as x approaches negative infinity”
OR as x gets very small
OR as we move forever to the left
“as x approaches positive infinity”
OR as x gets very large
OR as we move forever to the right
( )f x“f(x) approaches negative infinity”
OR y gets very small
OR the graph falls
“f(x) approaches positive infinity”
OR y gets very large
OR the graph rises
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End Behavior of Polynomial Functions
1
1 1 0( ) ...n n
n nf x a x a x a x a
• Leading term and degree tell end
behavior
• Follow these rules…
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End Behavior of Polynomial Functions
1
1 1 0( ) ...n n
n nf x a x a x a x a
0 and even,
( ) as and ( ) as
na n
f x x f x x
0 and even,
( ) as and ( ) as
na n
f x x f x x
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End Behavior of Polynomial Functions
1
1 1 0( ) ...n n
n nf x a x a x a x a
0 and odd,
( ) as and ( ) as
na n
f x x f x x
0 and odd,
( ) as and ( ) as
na n
f x x f x x
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End Behavior of Polynomial
Functions
Leading
Coefficient
Degree Left
Behavior
Right
Behavior
+ Even Rises Rises
+ Odd Falls Rises
- Even Falls Falls
- Odd Rises Falls
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End Behavior of Polynomial
Functions
Leading
Coefficient
Degree
+ Even
+ Odd
- Even
- Odd
as x
( )f x ( )f x
( )f x( )f x
( )f x ( )f x
( )f x ( )f x
as x
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Examples
• Describe the end behavior of the
function.3 21. ( ) 2 3f x x x x
Leading Coefficient: + Degree: Odd
( ) as
( ) as
f x x
f x x
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Assignment
• p.334
#53-64 all, 65-79 odds
(20 problems)
• #65-79 odds (draw sketches of graphs
using graphing calculator, trace the curve
to get fairly accurate graphs)