# 3.5 Higher – Degree Polynomial Functions and Graphs

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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. Whats the domain of a polynomial function?. - PowerPoint PPT Presentation

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3.5 Higher Degree Polynomial Functions and Graphs

3.5 Higher Degree Polynomial Functions and GraphsPolynomial FunctionDefinition: A polynomial function of degree n in the variable x is a function defined by

Where each ai(0 i n-1) is a real number, an 0, and n is a whole number. Whats the domain of a polynomial function?P(x) = anxn + an-1xn-1 + + a1x + a0Get to know a polynomial functionP(x) = anxn + an-1xn-1 + + a1x + a0an : Leading coefficientanxn : Dominating terma0 : Constant termCubic FunctionsP(x) = ax3 + bx2 + cx + d(b)(a)(d)(c)Quartic FunctionsP(x) = ax4 + bx3 + cx2 + dx + e(b)(a)(d)(c)ExtremaTurning 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 ExtremaLet 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.

ExampleLocal minimum pointLocal minimumpointLocal minimum &Absolute minimumpointLocal minimum pointLocal minimum pointA function can only have one and only one absolute minimum of maximumHidden behaviorHidden behavior of a polynomial function is the function behaviors which are not apparent in a particular window of the calculator. Number of Turning PointsThe 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) = x2 f(x) = x3End BehaviorDefinition: 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 BehaviorOdd degreea > 0a < 0Even degreea > 0

a < 0

exampleDetermining end behavior Given the Polynomial f(x) = x4 x2 +5x -4X 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) = x3 + 5x2 +5x -2Comprehensive GraphsA 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 behaviorexample1. f(x) = 2x3 x2 -22. f(x) = -2x3 - 14x2 + 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) = 0HomeworkPG. 210: 10-50(M5), 60, 63

KEY: 25, 60