Chapter 3: Polynomial Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs 3.3 Quadratic Equations and Inequalities 3.4 Further Applications of Quadratic Functions and Models 3.5 Higher-Degree Polynomial Functions and Graphs 3.6 Topics in the Theory of Polynomial Functions (I) 3.7 Topics in the Theory of Polynomial Functions (II) 3.8 Polynomial Equations and Inequalities; Further Applications and Models

3.5 Higher Degree Polynomial Functions and Graphs an is called the leading coefficient anxn is called the dominating term a0 is called the constant term P(0) = a0 is the y-intercept of the graph of P Polynomial Function A polynomial function of degree n in the variable x is a function defined by where each ai is real, an  0, and n is a whole number.

3.5 Cubic Functions: Odd Degree Polynomials The cubic function is a third degree polynomial of the form In general, the graph of a cubic function will resemble one of the following shapes.

3.5 Quartic Functions: Even Degree Polynomials The quartic function is a fourth degree polynomial of the form In general, the graph of a quartic function will resemble one of the following shapes. The dashed portions indicate irregular, but smooth, behavior.

3.5 Extrema Turning points – where the graph of a function changes from increasing to decreasing or vice versa Local maximum point – highest point or “peak” in an interval function values at these points are called local maxima Local minimum point – lowest point or “valley” in an interval function values at these points are called local minima Extrema – plural of extremum, includes all local maxima and local minima

3.5 Extrema

3.5 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 a local maximum if P(c) > P(x) when x is near c. (d) P(c) is a local minimum if P(c) < P(x) when x is near c. The expression “near c” means that there is an open interval in the domain of P conaining c, where P(c) satifies the inequality.

3.5 Identifying Local and Absolute Extrema Example Consider the following graph. Identify and classify the local extreme points of f. Identify and classify the local extreme points of g. Describe the absolute extreme points for f and g. Local Min points: (a,b),(e,h); Local Max point: (c,d) Local Min point: (j,k); Local Max point: (m,n) f has an absolute minimum value of h, but no absolute maximum. g has no absolute extrema.

3.5 Number of Local Extrema A linear function has degree 1 and no local extrema. A quadratic function has degree 2 with one extreme point. A cubic function has degree 3 with at most two local extrema. A quartic function has degree 4 with at most three local extrema. Extending this idea: 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.

3.5 End Behavior Let axn be the dominating term of a polynomial function P. n odd If a > 0, the graph of P falls on the left and rises on the right. If a < 0, the graph of P rises on the left and falls on the right. n even If a > 0, the graph of P opens up. If a < 0, the graph of P opens down.

3.5 Determining End Behavior Match each function with its graph. Solution: B. A. C. D. f matches C, g matches A, h matches B, k matches D.

3.5 x-Intercepts (Real Zeros) Number of x-Intercepts (Real Zeros) of a Polynomial Function The graph of a polynomial function of degree n will have at most n x-intercepts (real zeros). Example Find the x-intercepts of Solution By using the graphing calculator in a standard viewing window, the x-intercepts (real zeros) are –2, approximately –3.30, and approximately 0.30.

3.5 Analyzing a Polynomial Function Determine its domain. Determine its range. Use its graph to find approximations of local extrema. Use its graph to find the approximate and/or exact x- intercepts. Solution Since P is a polynomial, its domain is (–, ). Because it is of odd degree, its range is (–, ).

3.5 Analyzing a Polynomial Function Two extreme points that we approximate using a graphing calculator: local maximum point (– 2.02,10.01), and local minimum point (0.41, – 4.24). Looking Ahead to Calculus The derivative gives the slope of f at any value in the domain. The slope at local extrema is 0 since the tangent line is horizontal.

3.5 Analyzing a Polynomial Function (d) We use calculator methods to find that the x-intercepts are –1 (exact), 1.14(approximate), and –2.52 (approximate).

3.5 Comprehensive Graphs The most important features of the graph of a polynomial function are: intercepts, extrema, end behavior. A comprehensive graph of a polynomial function will exhibit the following features: all x-intercepts (if any), the y-intercept, all extreme points (if any), enough of the graph to reveal the correct end behavior.

3.5 Determining the Appropriate Graphing Window The window [–1.25,1.25] by [–400,50] is used in the following graph. Is this a comprehensive graph? Solution Since P is a sixth degree polynomial, it can have up to 6 x-intercepts. Try a window of [-8,8] by [-1000,600].