Warm Up What do you know about the graph of f(x) = (x – 2)(x – 4)2 ?

Analyzing Graphs of Polynomial Functions 5.4 Analyzing Graphs of Polynomial Functions

Zeros, Factors, Solutions, and Intercepts Let f(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x1 + a0 be a polynomial function. The following statements are equivalent. Zero: k is a zero of the polynomial function f. Factor: x – k is a factor of the polynomial f(x). Solution: k is a solution of the polynomial equation f(x) = 0. x-intercept: k (if it is real) is an x-intercept of the graph of the polynomial function f.

Find approximate real solutions for polynomial equations by graphing 1. Put polynomial equation into standard form. 2. Set equal to y 3. Graph 4. Find the zeros/ real solutions table – y-value of 0 calc – zero

Example 1: x3 + 6x2 – 4x – 24 = 0 b. x4 – 9x2 = 0

Example 2 Determine consecutive values of x between which each real zero of the function f(x) = x4 – x3 – 4x2 + 1 is located using the table below.

To identify key points on a graph of a polynomial function Turning points are called local maximums and local minimums because they are the highs or lows for the area. If there are two turning points without a zero between them  there are imaginary zeros.

Example 3: Estimate the coordinates of each turning point and state whether each corresponds to a local max or local min. Then list all real zeros and determine the least degree the function can have.

Example 4 Graph each function. Identify the x-intercepts (zeros), local maximums, and local minimums. a. f(x) = x3 + 2x2 – 5x + 1 x y

b. f(x) = 2x4 – 5x3 – 4x2 – 6 x y

Example 5 The weight w, in pounds, of a patient during a 7-week illness is modeled by the function w(n) = 0.1n3 – 0.6n2 + 110, where n is the number of weeks since the patient became ill. Graph the equation by making a table of values for weeks 0 through 7. Plot the points and connect with a smooth curve. Describe the turning points of the graph and its end behavior.

There is a relative minimum at week 4. w(n) → ∞ as n → ∞.