Section 3.2
Polynomial Functions
and Their Graphs
JMerrill 2005
Revised 2008
What is a polynomial?
• An expression in the form of
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao
where n is a non-negative integer and a2, a1, and a0 are real
numbers.
• The function is called a polynomial function of x with degree n.
• A polynomial is a monomial or a sum of terms that are monomials
• Polynomials can NEVER have a negative exponent or a variable
in the denominator!
• The term containing the highest power of x is called the leading
coefficient, and the power of x contained in the leading terms is
called the degree of the polynomial.
Significant features
• The graphs of polynomial functions are
continuous (no breaks—you draw the entire
graph without lifting your pencil). This is opposed
to discontinuous functions (remember piecewise
functions?).
• This data is continuous as opposed to discrete.
Examples of Polynomials
Degree
Name
Example
0
Constant
5
1
Linear
3x+2
2
Quadratic
X2 – 4
3
Cubic
X3 + 3x + 1
4
Quartic
-3x4 + 4
5
Quintic
X5 + 5x4 - 7
Significant features
• The graph of a polynomial function has only
smooth turns. A function of degree n has at most
n – 1 turns.
− A 2nd degree polynomial has 1 turn
− A 3rd degree polynomial has 2 turns
− A 5th degree polynomial has…
Cubic Parent Function
Draw the parent
functions on the
graphs.
X
Y
-3
-27
-2
-8
f(x) = x3
-1
-1
Domain  ,  
0
0
1
1
2
8
3
27
Range  -, 
Quartic Parent Function
Draw the parent
functions on the
graphs.
X
Y
-3
81
-2
16
f(x) = x4
-1
1
Domain  ,  
0
0
Range 0, 
1
1
2
16
3
81
Graph and Translate
Start with the graph of y = x3. Stretch it by a factor of 2 in
the y direction. Translate it 3 units to the right.
X
Y
-3 y-27
Equation

-2
-8
2  x  3
Domain-1  -1
, 
0
0
Range1  1,  
2
8
3
27
3
X
Y
0
-54
1
-16
2
-2
3
0
4
2
5
16
6
54
Graph and Translate
Start with the graph of y = x4. Reflect it across
the x-axis. Translate it 2 units down.
Y
X
Y
-81
-3
-83
-16
-2
-18
-1
-1
-3
Range
1
1  , 21
0
0
-2
-1
1
-3
X
Y
-3
81
-2
16
X
Equation y   x 4  2-3
-2
Domain

-1
1  , -1
0
0
0
2
16
2
-16
2
-18
3
81
3
-81
3
-83
Max/Min
• A parabola has a
maximum or a minimum
• Any other polynomial
function has a local max or
a local min. (extrema)
Local
max
min
max
Local
min
Leading Coefficient Test
• As x moves without bound to the left or right, the
graph of a polynomial function eventually rises or
falls like this:
• In an odd degree polynomial:
− If the leading coefficient is positive, the graph falls
to the left and rises on the right
− If the leading coefficient is negative, the graph rises
to the left and falls on the right
• In an even degree polynomial:
− If the leading coefficient is positive, the graph rises
on the left and right
− If the leading coefficient is negative, the graph falls
to the left and right
End Behavior
• If the leading coefficient of a polynomial function
is positive, the graph rises to the right.
y = x2
y = x3 + …
y = x5 + …
Finding Zeros of a Function
• If f is a polynomial function and a is a real number,
the following statements are equivalent:
• x = a is a zero of the function
• x = a is a solution of the polynomial equation f(x)=0
• (x-a) is a factor of f(x)
• (a,0) is an x-intercept of f
Example
• Find all zeros of f(x)=x3 – x2 – 2x
• Set function = 0
0 = x3 – x2 – 2x
• Factor
0 = x(x2 – x – 2)
• Factor completely
0 = x(x – 2)(x + 1)
• Set each factor = 0, solve
0=x
0 = x – 2; so x = 2
0 = x + 1; so x = -1
You Do
• f(x)=-2x4 + 2x2
• Degree of polynomial?
− Even
• End behavior?
− Falls to the left and falls to the right
• Zeros?
− X = 0, 1, -1
Multiplicity (repeated zeros)
• How many roots?
• How many roots?
3 is a
double root
3 roots; x = 1, 3, 3.
3 is a
double root
4 roots; x = 1, 3, 3, 4.
Roots of Polynomials
• How many roots?
Double
Double
roots
roots
5 roots: x = 0, 0, 1, 3, 3.
0 and 3 are double roots
Triple
root –
• How many roots? lies flat
then
crosses
axis
3 roots; x = 2, 2, 2