POLYNOMIAL FUNCTIONS - andrews.edu
Transcript of POLYNOMIAL FUNCTIONS - andrews.edu
Adding and subtracting polynomials
Combine like terms
3(2x + 3) + 4(x – 2) (4x2 + x - 5) - (2x3 + 3x2 - 4x + 2)
Pg. 151 #1; pg. 152 #1-10
3(2x + 3) + 4(x – 2)6x + 9 + 4x – 8 10x – 1
(4x2 + x – 5) – (2x3 + 3x2 – 4x + 2)4x2 + x – 5 – 2x3 – 3x2 + 4x – 2- 2x3 + x2 + 5x – 7
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Transum Mathematics
Ans: 2n + 6Possible pairs: 2(n+3); -2(-n-3); 6(1/3n+1); etcMatthew 18 Parable of the lost sheep
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4.1 GRAPHING POLYNOMIAL FUNCTIONSObjectives:
I can identify and evaluate polynomial functions.
I can graph polynomial functions.
I can describe end behavior of polynomial functions.
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Work with a partner
Use a calculator and graph each function
Describe the end behavior (What is happening
at the left and right ends of the graph?)
Identify the term with the greatest exponent.
How does the exponent affect the graph?
𝑓 𝑥 = 𝑥 + 1
g 𝑥 = 3𝑥2 + 4𝑥 + 1
h 𝑥 = −𝑥3 − 1
𝑖 𝑥 = −𝑥4 + 3𝑥2 + 2
F(x): falls, risesG(x): rises, risesH(x): rises, fallsI(x): falls, falls
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Polynomial Function
Of the form f(x) = an xn + an-1 xn-1 + ⋅ ⋅ ⋅ + a1 x + a0
an ≠ 0; exponents are whole numbers (positive); coefficients are real.
Polynomial in one variable
Function that has one variable and there are powers of that
variable and all the powers are positive.
Degree
Highest power of the variable.
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Is the function a polynomial in one variable? If so, what is the degree?
2𝑥3 + 5𝑥 + 8
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𝑥
−47𝑥5932 − 32𝑥4278 + 1
4𝑥12 + 3𝑥𝑦 + 2
2𝑥3 + 5𝑥 + 8 yes; 3
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𝑥no
−47𝑥5932 − 32𝑥4278 + 1 yes, 5932
4𝑥12 + 3𝑥𝑦 + 2 no
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Common Types of Polynomial Functions
Degree Type Example
0 Constant 𝑓 𝑥 = 1
1 Linear 𝑓 𝑥 = 3𝑥 + 2
2 Quadratic 𝑓 𝑥 = 2𝑥2 − 𝑥 + 3
3 Cubic 𝑓 𝑥 = 𝑥3 + 𝑥2 + 4𝑥 − 1
4 Quartic 𝑓 𝑥 = −5𝑥4 + 2𝑥 + 3
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Direct Substitution
Plug in for all x’s
Evaluate 𝑓 𝑥 = 2𝑥3 + 3𝑥2 − 5𝑥 + 8 when x = 2
Replace all x’s with 2F(2) = 2(2)^3 + 3(2)^2 - 5(2) + 8= 2(8) + 3(4) – 10 + 8= 16 + 12 – 10 +8= 26
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
End Behavior
The behavior of the graph as x approaches positive infinity or negative
infinity.
Determined by the degree and sign of the leading coefficient.
Write: 𝑓 𝑥 → −∞ 𝑎𝑠 𝑥 → −∞ 𝑎𝑛𝑑𝑓 𝑥 → −∞ 𝑎𝑠 𝑥 → +∞
Leading Coefficient + Leading Coefficient -
Even Degree
Odd Degree
Even, negative
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Graphing
Make a table of values
Plot the points
Draw the curve
Check end behaviors
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4.1 GRAPHING POLYNOMIAL FUNCTIONS
Graph 𝑓 𝑥 = 𝑥3 + 𝑥2 − 4𝑥 + 2
Pg. 158 # 1, 5, 7, 9, 13, 15, 19, 21, 23, 25, 29,
31, 33, 35, 39, 51, 53, 55, 57, 59
Points (-2,6), (-1,6), (0,2), (1,0), (2,6)
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