Section 2.3
Polynomial Functions & their Graphs
Objectives
– Identify polynomial functions.
– Recognize characteristics of graphs of
polynomials.
– Determine end behavior.
– Use factoring to find zeros of polynomials.
– Identify zeros & their multiplicities.
– Use Intermediate Value Theorem.
– Understand relationship between degree &
turning points.
– Graph polynomial functions.
DEFINITIONS -- Review
• Monomial – an expression that is one number,
one variable, or a product of a number & one or
more variables (one term)
• Binomial = 2 terms
• Trinomial = 3 terms
• Polynomial – a monomial or sum or difference of
two or more monomials (one or more terms)
• Write in decreasing power of the variable.
• Examples: 3xy
2y – 4
x2 + 3x + 2
z3 – 3z2 + 7z - 8
Review -- Finding the Degree
• The degree of a polynomial is the highest degree
of any one term. (Look for highest exponent. If
more than one variable in a term, add the
exponents.)
• Know the terms: constant, linear, quadratic,
cubic, & quartic. (next slide)
• Clue: The degree tells you what type of graph
the equation will be and the maximum number
of “zeros” (solutions) of the polynomial.
Polynomial Function
Degree
Example
Constant
0
f(x) = 4
Linear
1
f(x) = 3x + 1
Quadratic
2
f(x) = 4x2  x + 9
Cubic
3
f(x) = x3 +2x2  x + 11
Quartic
4
f(x) = x4  3.2x3 + 0.1x
Polynomial Functions
A polynomial of degree n is a function of the form
f(x) = anxn + an-1xn-1 + ... + a1x + a0
where an  0.
• coefficients: numbers an, an-1,…a2, a1, a0
• constant coefficient or constant term: a0
• coefficient of the highest power: an; a.k.a. the
leading coefficient
• leading term:
anxn
Graphs of Polynomials
• Degree 0 or 1
Graphs are lines
– ex. f(x) = 3 or f(x) = x –
5
• Degree 2
Graphs are parabolas
– ex. f(x) = x2 + 4x + 8
• Degree greater than 2
Graphs smooth curve
– ex. f(x) = x3
• These graphs are
continuous & will not
have the following:
– Break or hole
– Corner or cusp
The domain of these graphs is the set of all real numbers.
GRAPHS - What similarities do you see?
Leading Term Test, page 290
• End behavior – a description of what happens as
x approaches infinity and negative infinity
(becomes very large & very small)
Note: End behavior is determined by the
leading term (when written in descending
order)
Find the degree of the polynomial and the
sign of the leading term’s coefficient.
Leading Coefficient Test
(a.k.a. Leading-Term Test)
Match each of the following functions with one of
the graphs AD, which follow.
1)
3)
f ( x)  3x 4  2 x 3  3
f ( x)  x5  14 x  1
2)
4)
f ( x)  5 x3  x 2  4 x  2
f ( x)   x 6  x 5  4 x 3
Graphs
Solution
Leading Term
Degree of
Sign of
Leading Term Leading Coeff.
Graph
3x4
Even
Positive
D
5x3
Odd
Negative
B
x5
Odd
Positive
A
x6
Even
Negative
C
Even- & Odd-Degree Functions
Graphing Calculators
• See Example 3, page 291.
Don’t let your viewing window fool you.
• Answer Check Point 3, page 292.
Zeros of Polynomials
• Solutions or Roots
• Real Zeros are also x-intercepts of
the graph
• Zeros of the function: find by solving
f(x)=0. Reminder: You may need to
factor or use Q.F. to find zeros.
• Note: A polynomial of degree n will
have at least one solution and at most
n solutions.
Example
• Find the zeros of f(x) = (x + 5)3(x – 4)(x + 1)2.
See Example 4, page 292.
• Check Point 4: Find all zeros of
f(x) = x3 + 2x2 – 4x – 8.
See Example 5, page 293.
• Check Point 5: Find all zeros of
f(x) = x4 -4x2.
Multiplicity - the number of times a
factor appears
If (x - c)k, k > 1, is a factor of a polynomial function &
K is odd
– The graph crosses the
x-axis at (c, 0)
K is even
– The graph is tangent
to the x-axis at (c, 0)
Multiplicity Example
If f(x) = (x - 4)3,
K is odd
– The graph crosses the
x-axis at (4, 0)
If f(x) = (x –2)2 ,
K is even
– The graph is tangent
to the x-axis at (c, 0)
Find the roots and multiplicity of
y = (x + 2)²(x − 1)³
Answer.
−2 is a root of multiplicity 2,
and 1 is a root of multiplicity 3.
See Example 6, page 294.
• Check Point 6: Find the zeros of
f(x) = -4(x + ½)2(x - 5)3 and give the
multiplicity of each zero. State whether the
graph crosses the x-axis or touches the x-axis
and turns around at each zero.
Intermediate Value Theorem, pg. 294
• For any polynomial function P(x) with real
coefficients, suppose that for a  b, P(a) and
P(b) are of opposite signs. Then the function
has at least one real zero between a and b.
(The graph must cross the x-axis)
The Intermediate Value
Theorem
In other words, if one point is above
the x-axis and the other point is
below the x-axis, then because f(x)
is continuous and will have to cross
the x-axis to connect the two
points, f(x) must have a zero
somewhere between a and b.
See Example 7.
• Check Point 7: Show that the polynomial
function f(x) = 3x3 – 10x + 9 has a real zero
between -3 and -2.
Turning points of a polynomial
• If a polynomial is of degree “n”, then it has at
most n - 1 turning points.
• A graph changes direction at a turning point.
Graphing a Polynomial Function, pg.
295
1.
Use the leading term to determine the end behavior.
2.
Find all its real zeros (x-intercepts). Set y = 0.
3.
Use the x-intercepts to divide the graph into
intervals and choose a test point in each interval to
graph.
4.
Find the y-intercept. Set x = 0.
5.
Use any additional information (i.e. symmetry,
turning points or multiplicity) to graph the function.
Graph, state zeros & end behavior
f ( x)  2 x 3  12 x 2  18 x  2 x( x 2  6 x  9)
f ( x)  2 x( x  3) 2
• END behavior: 3rd degree equation and the leading coefficient is
negative,
(f(x) goes UP as you move to the left.)
(f(x) goes DOWN as you move to the right.)
• ZEROS: x = 0, x = 3 of multiplicity 2
• Graph on next page
Graph f(x).
f ( x)  2 x3  12 x 2  18 x
•f(x) goes UP as you move left
•f(x) goes DOWN as you
move right
•ZEROS: x = 0, x = 3 of
multiplicity 2
4 – 4x3 + 3x2
Graphing Example g(x)
=
x
y
10
8
6
4
2
10
8
6
4
2
2
2
4
6
8
10
4
6
8
10
x