2Higher Degree Polynomial Functions and Graphs an is called the leading coefficientn is the degree of the polynomiala0 is called the constant termPolynomial FunctionA polynomial function of degree n in the variable x is a function defined bywhere each ai is real, an 0, and n is a whole number.
3Polynomial Function in General Form Polynomial FunctionsPolynomial Function in General FormDegreeName of Function1Linear2Quadratic3Cubic4QuarticTeachers: This definition for ‘degree’ has been simplified intentionally to help students understand the concept quickly and easily.The largest exponent within the polynomial determines the degree of the polynomial.
4Maximum Number of Zeros: 0 Polynomial Functionsf(x) = 3ConstantFunctionDegree = 0Maximum Number of Zeros: 0
5Maximum Number of Zeros: 1 Polynomial Functionsf(x) = x + 2LinearFunctionDegree = 1Maximum Number of Zeros: 1
6Maximum Number of Zeros: 2 Polynomial Functionsf(x) = x2 + 3x + 2QuadraticFunctionDegree = 2Maximum Number of Zeros: 2
7Maximum Number of Zeros: 3 Polynomial Functionsf(x) = x3 + 4x2 + 2Cubic FunctionDegree = 3Maximum Number of Zeros: 3
8Maximum Number of Zeros: 4 Polynomial FunctionsQuartic FunctionDegree = 4Maximum Number of Zeros: 4
9Leading CoefficientThe leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.For example, the quartic functionf(x) = -2x4 + x3 – 5x2 – 10 has a leading coefficient of -2.
10The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial functionf (x) = anxn + an-1xn-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)eventually rises or falls. In particular,For n odd: an > an < 0If the leading coefficient is positive, the graph falls to the left and rises to the right.If the leading coefficient is negative, the graph rises to the left and falls to the right.Rises rightFalls leftFalls rightRises left
11The Leading Coefficient Test As x increases or decreases without bound, the graph of the polynomial functionf (x) = anxn + an-1xn-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)eventually rises or falls. In particular,For n even: an > an < 0If the leading coefficient is positive, the graph rises to the left and to the right.If the leading coefficient is negative, the graph falls to the left and to the right.Rises rightRises leftFalls leftFalls right
12ExampleUse the Leading Coefficient Test to determine the end behavior of the graph of f (x) = x3 + 3x2 - x - 3.Falls leftyRises rightx
13Determining End Behavior Match each function with its graph.B.A.C.D.
14Quartic PolynomialsLook at the two graphs and discuss the questions given below.Graph BGraph A1. How can you check to see if both graphs are functions?2. How many x-intercepts do graphs A & B have?3. What is the end behavior for each graph?4. Which graph do you think has a positive leading coeffient? Why?5. Which graph do you think has a negative leading coefficient? Why?
15x-Intercepts (Real Zeros) Number Of x-Intercepts of a Polynomial FunctionA polynomial function of degree n will have a maximum of n x- intercepts (real zeros).Find all zeros of f (x) = -x4 + 4x3 - 4x2.-x4 + 4x3 - 4x2 = We now have a polynomial equation.x4 - 4x3 + 4x2 = Multiply both sides by -1. (optional step)x2(x2 - 4x + 4) = Factor out x2.x2(x - 2)2 = Factor completely.x2 = or (x - 2)2 = Set each factor equal to zero.x = x = Solve for x.(0,0) (2,0)
16Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.
17Example Find the x-intercepts and multiplicity of f(x) =2(x+2)2(x-3) Zeros are at(-2,0)(3,0)
18ExtremaTurning points – where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n 1 is at most n – 1.Relative maximum point – highest point or “peak” in an intervalfunction values at these points are called local maximaRelative minimum point – lowest point or “valley” in an intervalfunction values at these points are called local minimaExtrema – plural of extremum, includes all relativel maxima and local minima
20Number of Relative Extrema A linear function has degree 1 and no relative extrema.A quadratic function has degree 2 with one relative extreme point.A cubic function has degree 3 with at most two relative extrema.A quartic function has degree 4 with at most three relative extrema.How does this relate to the number of turning points?
21Comprehensive Graphs intercepts, extrema, end behavior. 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 exhibit end behavior.