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
g ( x )  x
h (x )  x
2
4
negative leading
coefficient and
even degree
as x  , f (x )  
6
as x  , f (x )  
Graph each of the following:
g (x )  x
3
positive leading
coefficient and
odd degree
h (x )  x
5
as x  , f (x )  
as x  , f (x )  
f (x )  x
Graph each of the following:
f (x )  x
g (x )  x
3
h (x )  x
5
negative leading
coefficient and
odd degree
as x  , f (x )  
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
f (x )  x  x  20x
4
3
• Find the x-intercept(s).
• Find the y-intercept(s).
• Describe the end behaviors.
2
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 inches.
Consider the function:
f (x )  2x  3x  120x  10
3
2
with  5  x  5.
Find the absolute maximum
and absolute minimum of
the graph.