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3.5 Higher – Degree Polynomial Functions and Graphs
Polynomial Function Definition: A polynomial function of degree n in the variable x is a function defined by Where each a i (0 ≤ i ≤ n-1) is a real number, a n ≠ 0, and n is a whole number. What’s the domain of a polynomial function? P(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0
Get to know a polynomial function P(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 a n : Leading coefficient a n x n : Dominating term a 0 : Constant term
Extrema Turning points: points where the function changes from increasing to decreasing or vice versa. Local maximum point: the highest point at a peak. The corresponding function values are called local maxima. Local minimum point: the lowest point at a valley. The corresponding function values are called local minima. Extrema: either local maxima or local minima.
Absolute and Local Extrema Let c be in the domain of P. Then (a) P(c) is an absolute maximum if P(c) ≥ P(x) for all x in the domain of P. (b) P(c) is an absolute minimum if P(c) ≤ P(x) for all x in the domain of P. (c) P(c) is an local maximum if P(c) ≥ P(x) when x is near c. (d) P(c) is an local minimum if P(c) ≤ P(x) when x is near c.
Example Local minimum point Local minimum point Local minimum & Absolute minimum point Local minimum point Local minimum point A function can only have one and only one absolute minimum of maximum
Hidden behavior Hidden behavior of a polynomial function is the function behaviors which are not apparent in a particular window of the calculator.
Number of Turning Points The number of turning points of the graph of a polynomial function of degree n ≥ 1 is at most n – 1. Example: f(x) = x f(x) = x 2 f(x) = x 3
End Behavior Definition: The end behavior of a polynomial function is the increasing of decreasing property of the function when its independent variable reaches to ∞ or - ∞ The end behavior of the graph of a polynomial function is determined by the sign of the leading coefficient and the parity of the degree.
End Behavior Odd degree a > 0 a < 0 Even degree a > 0 a < 0
example Determining end behavior Given the Polynomial f(x) = x 4 –x 2 +5x -4
X – Intercepts (Real Zeros) Theorem: The graph of a polynomial function of degree n will have at most n x-intercepts (real zeros). Example: P(x) = x 3 + 5x 2 +5x -2
Comprehensive Graphs A comprehensive graph of a polynomial function will exhibit the following features: 1. all x-intercept (if any) 2. the y-intercept 3. all extreme points(if any) 4. enough of the graph to reveal the correct end behavior
example 1. f(x) = 2x 3 – x 2 -2 2. f(x) = -2x 3 - 14x 2 + 2x + 84 a) what is the degree? b) Describe the end behavior of the graph. c) What is the y-intercept? d) Find any local/absolute maximum value(s).... local/absolute maximum points. [repeat for minimums] e) Approximate any values of x for which f(x) = 0