Section 5.1 Polynomial Functions and Models
Objectives: Identify polynomial functions and their degree;
graph using transformations; identify zeros and their
multiplicity; analyze the graph of a polynomial function.
A polynomial function is a sum of terms Ax n where A are real
numbers and n is a nonnegative integer.
 The domain is the set of all real numbers.
 The degree is the largest power of x.
 The zero polynomial function f  x   0 is not assigned a
degree.
 The constant polynomial function f  x   b has degree 0
since bx 0  b  1  b .
Work #1 – 4
A power function of degree n is a monomial of the form
f  x   ax n where n > 0 and n is an integer. For example,
given the polynomial f  x   6 x 4  3 x 3  7 , the power function
is f  x   6x 4 .
5.1 - 2
Graphing Polynomials
We will use the end behavior of the power function to
determine what happens to a polynomial graph as x becomes
very large or very small.
 If the degree of the polynomial is odd, the ends will point
in opposite directions.
o Positive leading coefficient: down on left / up on right
o Negative leading coefficient: up on left / down on right
 If the degree of the polynomial is even, the ends will point
in the same direction.
o Positive leading coefficient: up on left / up on right
o Negative leading coefficient: down left / down right
The graph of every polynomial function is both smooth (no
sharp corners) and continuous (no gaps or holes).
If f is a function and r is a real number for which f r   0 ,
then r is called a real zero of f. Therefore, the following are
equivalent:
 r is a real zero of a polynomial function f (real roots)
 r is an x-intercept of the graph of f
  x  r  is a factor of f
Work #5 – 7
5.1 - 3
Zeros and Their Multiplicities
If f is completely factored with factors  x  r m then r is a
multiple root that occurs m times. For example, in the
polynomial 0   x  54  x  63  x  3 , 5 is a zero with
multiplicity 4, 6 is a zero with multiplicity 3, and 3 is a zero
with multiplicity 1.
 If m is odd the graph of f crosses the x-axis at r.
 If m is even the graph of f touches the x-axis at r.
 The larger the value of m, the flatter (more compressed)
the graph becomes near r.
 Look for the degree of the polynomial—that is the most
real zeros you will have.
Work #8 – 11
Behavior Near a Zero
To determine behavior near a zero, keep the factor that gave
rise to the zero, and let x  r in the remaining factors.
Turning Points
The points where a graph changes direction are called turning
points – these are the local maximums and local minimums.
 If f is a polynomial function of degree n, then f has at
most n – 1 turning points.
Work #12 - 15