Pre-Calculus Chapter 2 Polynomial and Rational Functions
Warm Up 2.2 • A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals. Write the area A of the corral as a function of x. Find the dimensions that will produce the maximum area. y x x
2.2 Polynomial Functions of Higher Degree Objectives: • Use transformations to sketch graphs of polynomial functions. • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. • Find and use zeros of polynomial functions as sketching aids. • Use the Intermediate Value Theorem to help locate zeros of polynomial functions.
Vocabulary • Continuous Function • Polynomial Function • Monomial, Binomial, Trinomial Functions • Leading Coefficient Test • Zeros of Polynomial Functions • Repeated Zero • Multiplicity • Intermediate Value Theorem
Continuous Function • No breaks, holes or gaps. • Continuous or not?
Polynomial Function Characteristics • Continuous • Smooth, rounded turns. No sharp corners. • Polynomial or not?
Definition of a Polynomial Function • Polynomial Function of x with degree n f (x) = anxn + an – 1xn – 1 + … + a2x2 + a1x + a0 where: • n is a non-negative integer and • an, an – 1, … , a2, a1, a0 are real numbers • an ≠ 0
Monomial Functions • Simplest type of polynomial. • Monomial form: f (x) = xn, where n is an integer greater than zero. • As n increases, the graph gets flatter at the origin.
Transformations of Monomials • Sketch the graph of each function.
Leading Coefficient Test • Using the degree of the function (n) and the leading coefficient (an), we can determine the end behavior of a function.
Use the Leading Coefficient Test • Describe the end behavior of each function.
Zeros of Polynomial Functions • Zero = Root = x-intercept = Solution • A number x for which f (x) = 0. • For a polynomial function f of degree n: • The function f has at most nreal zeros. • The graph of f has at most n – 1 relative extrema (relative minima or maxima).
Real Zeros of Polynomials • If f is a polynomial function and a is a real number, then the following statements are equivalent. • x = ais a zero of the function. • x = ais a solution of the polynomial equation f (x) = 0. • (x – a) is a factor of the polynomial f (x). • (a, 0) is an x-intercept of the graph of f.
Finding Zeros of Polynomials • Find all real zeros and relative extrema of the polynomial functions.
Warm Up 2.2.2 • Use your graphing calculator to find the zeros and relative extrema of the function
Repeated Zeros • A repeated zero of multiplicity koccurs if a function has a factor of the form (x – a)k, where k > 1.
Example - Multiplicity • Find all the real zeros of each function and determine the multiplicity of each zero. Solve algebraically and graphically.
Find a Function Given the Zeros • Find a polynomial function that has the given zeros.
Basic Curve Sketching • Apply the Leading Coefficient Test. • Find the zeros of the polynomial. • Plot a few additional points. • Draw the graph.
Curve Sketching Example 1 • Sketch the graph of by hand.
Curve Sketching Example 2 • Sketch the graph of by hand.
Homework 2.2 • Worksheet 2.2 # 47, 49, 53, 59, 61, 65, 67, 69, 71, 85
