This document focuses on the characteristics and behavior of polynomial functions, including the leading coefficient test, multiplicity of zeros, and the number of turning points. It provides instructions for activities related to identifying graphs of polynomial functions and describes how to analyze their behavior through various illustrative examples. Additionally, it discusses the continuous and smooth nature of polynomial graphs compared to non-polynomial graphs.

1. Graphs of Polynomial
Functions

2. Polynomial Functions
Objectives:
a. Describe the behavior of the graph using the
Leading Coefficient Test.
b. Identify the number of turning points and the
behavior of the graph based on the multiplicity of zeros.
c. Value accumulated knowledge as means of new
understanding.

3. INSTRUCTIONS:
1. Given the polynomial function,
match it to the graph on the
board.
2. The group who got the correct
answer earns 5 points.
FIND YOUR MATCH

4. These are some other things that we need
to take into consideration;
a. multiplicity of roots.
b. behavior of the graph
c. number of turning points
FIND YOUR MATCH

5. Illustrative examples:
1. Describe the behavior of the graph of f(𝑥) = (𝑥 + 1)2
(𝑥 + 2)(𝑥 − 2)(𝑥 − 3).
A. X- AND Y-INTERCEPTS:
x-intercepts:−2,−1,−1,2, 3
y-intercept: 12
The graph will intersect the x-axis at (−2,0),(−1,0),(2,0), (3,0) and the y-axis at (0,12).
B. MULTIPLICITY
If 𝑟 is a zero of odd multiplicity, the graph of (𝑥) crosses the x-axis at
r.
If 𝑟 is a zero of even multiplicity, the graph of (𝑥) is tangent to the x-axis at 𝑟.
Since the root -1 is of even multiplicity 2, then the graph of the polynomial is
tangent to the x-axis at -1.

6. Illustrative examples:
C. BEHAVIOR OF THE GRAPH:
The following characteristics of polynomial functions will give us additional information.
The graph of a polynomial function:
i. comes down from the extreme left and goes up to the extreme right if n is even and
> 0
𝑎𝑛
ii. comes up from the extreme left and goes up to the extreme right if n is odd and >
𝑎𝑛
0
iii. comes up from the extreme left and goes down to the extreme right if n is even and
< 0
𝑎𝑛
iv. comes down from the extreme left and goes down to the extreme right if n is odd and
< 0
𝑎𝑛
For additional help, we can summarize this in the figure:
n is even n is odd

7. End behavior - Polynomial functions always go towards  or - at either
end of the graph
Leading Coefficient (+) POSITIVE Leading Coefficient (-) NEGATIVE
Even Degree
Odd Degree
f(x) = x2 f(x) = -x2
f(x) = -x3
f(x) = x3
Illustrative examples:

8. Leading Coefficient Test
As x grows positively or negatively without bound, the value f(x) of the polynomial
function f(x) = an
xn
+ an – 1
xn – 1
+ … + a1
x + a0
(an
 0)
grows positively or negatively without bound depending upon the sign of the leading
coefficient an
and whether the degree n is odd or even.
x
y
x
y
n odd n even
an positive
an negative

9. Polynomial functions of the form f(x) = xn
, n  1 are called power functions.
If n is even,
their graphs
resemble the
graph of
f(x) = x2
.
If n is odd, their graphs
resemble the graph of
f(x) = x3
.
x
y
x
y
f(x) = x2
f(x) = x5
f(x) = x4
f(x) = x3

10. Graphs of polynomial functions are continuous. That is, they have no breaks, holes,
or gaps.
Polynomial functions are also smooth with rounded turns. Graphs with points or cusps
are not graphs of polynomial functions.
x
y
x
y
continuous
not
continuous
continuous
smooth not smooth
polynomial
not polynomial
not polynomial
x
y
f(x) = x3
– 5x2
+ 4x + 4

11. Example: Determine the multiplicity of the zeros
of f(x) = (x – 2)3
(x +1)4
.
Zero Multiplicity Behavior
2
–1
3
4
odd
even
crosses x-axis
at (2, 0)
touches x-axis
at (–1, 0)
Repeated Zeros
If k is the largest integer for which (x – a)k
is a factor of f(x) and k > 1, then a
is a repeated zero of multiplicity k.
1. If k is odd the graph of f(x) crosses the x-axis at (a, 0).
2. If k is even the graph of f(x) touches, but does not cross
through, the x-axis at (a, 0).
x
y

12. Illustrative examples:
If the polynomialfunction
𝑃(𝑥) = (𝑥 + 1)2
(𝑥 + 2)(𝑥 − 2)(𝑥 − 3) is written in the standard form then we
have
𝑃(𝑥) = 𝑥5
− 𝑥4
− 9𝑥3
+ 𝑥2
+ 20𝑥 + 12
We can easily see that this is a 5th degree polynomial. Thus, 𝑛 is odd.
The leading term is 𝑥5
, 𝑎𝑛
= 1 and 𝑎𝑛
> 0.
Therefore the graph of the polynomial comes up from the extreme left and
goes up to the extreme right if n is odd and 𝑎𝑛 > 0

13. Illustrative examples:
D. NUMBER OF TURNING POINTS:
Remember that the number of turning points in the graph of a polynomial is strictly
less than the degree of the polynomial.
Also, we must note that;
i.Quartic Functions: have an odd number of turning points; at most 3 turning points
ii.Quintic functions: have an even number of turning points, at most 4 turning points
iii.The number of turning points is at most (𝑛 − 1)
For our graph to pass through the intercepts (−2,0), (2,0), (3,0) and tangent at (−1,0),
there will be 4 turning points.

14. Illustrative examples:
2. Describe the behavior graph of 𝑦 = 𝑥4
−
5𝑥2
+ 4

15. Illustrative examples:
2. Describe the behavior graph of 𝑦 = 𝑥4
− 5𝑥2
+ 4
a. leading term: ______
b. behavior of the graph: ____________________ ( 𝑛 is odd and 𝑎𝑛 > 0)
c. x-intercepts: ________ the polynomial in factored form
is 𝑦 = (𝑥 − 2)2
(𝑥 + 3)
d. multiplicity of roots:_____
e. y-intercept:_________
f. number of turning points:

16. APPLICATION
1.Are the intercepts enough information for us to
graph polynomials?
2.How can we describe the behavior of the graph of
a polynomial function?
3.Is it possible for the degree of function to be less
than the number of turning points?

17. APPLICATION
Find the following then describe the behavior of the graph of 𝑝(𝑥) = 𝑥3
− 𝑥2
− 8𝑥 +
12
a. leading term: ______
b. behavior of the graph: ____________________ ( 𝑛 is odd and 𝑎𝑛 > 0)
c. x-intercepts: ________ the polynomial in factored form is 𝑦 = (𝑥 − 2)2
(𝑥 + 3)
d. multiplicity of roots:_____
e. y-intercept:_________
f. number of turning points:

18. INDIVIDUAL TASK
Describe the graph of the following polynomial
functions:
1. 𝑦 = 𝑥3
+ 3𝑥2
− 𝑥 − 3
2. 𝑦 = −𝑥3
+ 2𝑥2
+ 11𝑥 - 12

19. GROUP TASK
Describe the graph of the following
polynomial functions:
1. 𝑦 = 𝑥3
− 𝑥2
− 𝑥 + 1
2. 𝑦 = (2𝑥 + 3)(𝑥 − 1) (𝑥 − 4)

20. GENERALIZATION
Things to consider before we draw the graph of a polynomial function.
a. x- and y- intercepts
b. multiplicity of roots
If is a zero of odd multiplicity, the graph of ( ) crosses the x-axis at r.
𝑟 𝑥
If is a zero of even multiplicity, the graph of ( ) is tangent to the x-axis at .
𝑥 𝑟
c. behavior of the graph
The following characteristics of polynomial functions will give us additional information.
The graph of a polynomial function:
i. comes down from the extreme left and goes up to the extreme right if n is even and 𝑎𝑛 > 0
ii. comes up from the extreme left and goes up to the extreme right if n is odd and 𝑎𝑛 > 0
iii. comes up from the extreme left and goes down to the extreme right if n is even and 𝑎𝑛 < 0
iv. comes down from the extreme left and goes down to the extreme right if n is odd and 𝑎𝑛 < 0
For additional help, we can summarize this in the figure: n is even n is odd
an >0 an< 0

21. GENERALIZATION
d. number of turning points:
Remember that the number of turning points in the graph of a polynomial is
strictly less than the degree of the polynomial.
Also, we must note that;
i.Quartic Functions: have an odd number of turning points; at most 3 turning
points
ii.Quintic functions: have an even number of turning points, at most 4 turning
points
iii.The number of turning points is at most ( − 1)
𝑛

22. APPLICATION
For the given polynomial function
𝑦 = −(𝑥 + 2) (𝑥 + 1)4
(𝑥− 1)3
, describe or determine the following.
a. leading term
b. behavior of the graph
c. x-intercepts
d. multiplicity of roots e. y-intercept
f. number of turning points

23. For the given polynomial function 𝑦 = 𝑥6
+ 4𝑥5
+ 4𝑥4
− 2𝑥3
−5𝑥2
− 2𝑥,
describe or determine the following:
a. leading term
b. behavior of the graph
c. x-intercepts
d. multiplicity of roots
e. y-intercept
f. number of turning points
g. sketch
I WANT MORE!

24. Study:
Graph of the function
𝑦 = 𝑥6
+ 4𝑥5
+ 4𝑥4
− 2𝑥3
−5𝑥2
− 2𝑥
HOMEWORK