1. Polynomial Functions and Models
Section 4.1
Polynomial Functions and Models
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2. Identify polynomial functions and their degree.
Objectives
Identify polynomial functions and their degree.
Graph polynomial functions using transformations.
Identify the real zeros of a polynomial function and their multiplicity.
Analyze the graph of a polynomial function.
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7. Properties of Power Functions f(x)=xn where n is an even integer:
Symmetric with respect to y-axis.
D: (‒∞,∞) and R: [0,∞)
Always contains (-1,1), (0,0), and (1,1).
As n increases, in magnitude, the graph increases more rapidly when x<-1 or x>1, but near the origin, the graph flattens.
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8. Properties of Power Functions f(x)=xn where n is an odd integer:
Symmetric with respect to origin.
D and R: (‒∞,∞)
Always contains (-1,‒1), (0,0), and (1,1).
As n increases, in magnitude, the graph increases more rapidly when x<-1 or x>1, but near the origin, the graph flattens.
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11. Find a polynomial of degree 3 whose zeros are -4, -2, and 3.
Use a graphing utility to verify your result.
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12. For the polynomial, list all zeros and their multiplicities.
2 is a zero of multiplicity 1 because the exponent on the factor x – 2 is 1.
–1 is a zero of multiplicity 3 because the exponent on the factor x + 1 is 3.
3 is a zero of multiplicity 4 because the exponent on the factor x – 3 is 4.
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14. Turning Points (changes from increasing ↔ decreasing):
If f is a polynomial function of degree n, f has at most n‒1 turning points.
If the graph of a polynomial function f has n‒1 turning points, the degree of f is at least n.
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15. y = 4(x - 2)
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20. Use this notation:
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27. The polynomial is degree 3 so the graph can turn at most 2 times.
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29. Homework
4.1: # 13, 14, 15, 19, 41,
47, 49, 57, 61, 65, 69, 73
(#69 and 73: steps 1-3,6,7)