1 Algebra 2: Section 6.2 Evaluating and Graphing Polynomial Functions (Day 1)
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1 Algebra 2: Section 6.2 Evaluating and Graphing Polynomial Functions (Day 1)
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3 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
Transcript of 1 Algebra 2: Section 6.2 Evaluating and Graphing Polynomial Functions (Day 1)
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Algebra 2: Section 6.2Evaluating and Graphing Polynomial
Functions(Day 1)
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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
11 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
na
0a
11 1 0( ) n n
n nf x a x a x a x a
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Classifying Polynomial Functions
• Classify based on highest power of x• Power of x (degree)
– Constant: Degree = 0: f(x) =
– Linear: Degree = Degree = 1: f(x) = f(x) =
– Quadratic: Degree = Degree = 2: f(x) =f(x) =
– Cubic: Degree = Degree = 3: f(x) =f(x) =
– Quartic: Degree = Degree = 4: f(x) =f(x) =
24 2 5x x 3 24 3 1x x
4 26 4 2 5x x x
2 3x
5
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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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Examples3 42. ( ) 0.8 5g x x x
Yes4 3( ) 0.8 5f x x x
Degree: 4Type: QuarticLeading Coefficient: 1
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33) ( ) 3xf x x
No, because it has an "x as the exponent.
Evaluating a Polynomial• Direct substitution – putting in the value
of x in place of x and solving
• Synthetic substitution – similar to synthetic division but the answer is just the last number of the problem.
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Direct Substitution
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5 43. ( ) 3 5 10 when 2f x x x x x
5 4( 2) 3( 2) ( 2) 5( 2) 10 f
( 2) 3( 32) (16) 5( 2) 10 f ( 2) 96 16 10 10 f
( 2) 92f
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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 column 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 10 2
36
7
1414
28
28
5651
102
92
Coefficients of x written in order
Missing power of x, zero coefficient!
Number you are dividing by goes in front
Drop 1st number down
**Multiply Across…..Add Down
ANSWER!
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3 24. (4) 5 4 1f x x x
5 1 4 14
520
21
8480
320321
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Homework• P.333
16 - 26 evens28 & 3038 - 46 evens
