# Basic Functions and Their Graphs - Arizona State University Polynomial Functions Section 2.3 Objectives Find the x-intercepts and y-intercept of a polynomial function. Describe the end behaviors of a polynomial function. Write the equation of a polynomial function given the zeros and a point on the function. Determine the minimal degree of a polynomial given its graph. Solve a word problem involving polynomial

function. Objectives Use a graphing utility to find a local maximum or local minimum of a polynomial function. Use a graphing utility to find the absolute maximum or absolute minumum of a polynomial function. Use a graphing utility to find the intersection points of the graphs of two polynomials.

Vocabulary polynomial function

degree leading coefficient end behavior repeated zero multiplicity local minimum local maximum absolute minimum absolute maximum Graph each of the following: f (x ) x

2 g(x ) x 4 h(x ) x 6 positive leading coefficient and even degree

as x , f (x ) as x , f (x ) Graph each of the following: f (x ) x 2 g(x ) x 4 h(x ) x

6 negative leading coefficient and even degree as x , f (x ) as x , f (x ) Graph each of the following: f (x ) x g(x ) x

3 h(x ) x 5 positive leading coefficient and odd degree as x , f (x ) as x , f (x )

Graph each of the following: f (x ) x g(x ) x h(x ) x 3 negative leading coefficient and odd degree as x , f (x )

5 as x , f (x ) For the function f (x ) (x 2)(x 3)(5x 4) Find the x-intercept(s). Find the y-intercept(s). Describe the end behaviors. For the function 4

3 f (x ) x x 20x 2 Find the x-intercept(s). Find the y-intercept(s). Describe the end behaviors. Find a possible formula for the polynomial of degree 4 that has a root of

multiplicity 2 at x = 2 and roots of multiplicity 1 at x = 0 and x = -2 that goes through the point (5, 63). What is the smallest possible degree of the polynomial whose graph is given below. A box without a lid is constructed from a 36 inch

by 36 inch piece of cardboard by cutting x inch squares from each corner and folding up the sides. Determine the volume of the box as a function of the variable x. Use a graphing utility to approximate the values of x that produce a volume of 3280.5 cubic Consider the function: 3

2 f (x ) 2x 3x 120x 10 with 5 x 5. Find the absolute maximum and absolute minimum of the graph.