10
Mathematics
Quarter 2 - Module 2
GRAPHING POLYNOMIAL FUNCTIONS
Department of Education ● Republic of the Philippines
Mathematics- Grade 10
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Quarter 2 - Module 2: GRAPHING POLYNOMIAL FUNCTIONS
First Edition, 2020
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Published by the Department of Education
Secretary: Leonor M. Briones
Development Team of the Module
Author:
Marjury R. Gallardo
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Rhodel A. Lamban, PhD
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Teodoro P. Casiano
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10
Mathematics
Quarter 2 - Module 2
GRAPHING POLYNOMIAL FUNCTIONS
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Table of Contents
What This Module is About
What I Need to Know
How to Learn from this Module
Icons of this Module
Lesson 1: The Leading Coefficient Test
What I Need to Know
What I Know
What’s In
What’s New
What Is It
What’s More
What I Have Learned
What I Can Do
Assessment
Additional Activities
1
1
3
4
5
9
11
12
12
14
Lesson 2: Graphing Polynomial Functions
What I Need to Know
What I Know
What’s In
What’s New
What Is It
What’s More
What I Have Learned
What I Can Do
Assessment
Additional Activities
16
16
18
19
19
23
23
24
25
26
Summary
27
Post-Test
28
Key to Answers
30
References
39
What This Module is About
This module has been designed to help both the teacher and the
learner make learning Mathematics as simple as possible. From its title, the
learner is expected to learn how to graph polynomial functions. The
requirement to understand this module easily is the learner’s knowledge on
polynomial equations.
Mathematics helps us think logically and reason out analytically. The
vision of this module is to help the learner understand further the idea of
functions. Functions is everywhere. For example, the amount of medicine that
a person should take depends on the age of a person. The medicine dose
changes when the age is changed.
Desmos graphing application and Microsoft M
Notes to the Teacher:
This module has been made with the thought of a teacher standing in
front of the class for discussions as well as the thought of one-on-one setting.
This module focuses in graphing polynomial functions in a step-by-step
process. May this help you in organizing your lessons and discussing your
lessons in detail.
Notes to the Learner:
This is your last year in Junior High School. You are now in Grade 10,
therefore it is expected that you have mastered the basic skills and foundation
in Mathematics for these are the weapons that will help you in this higher
order thinking skills battlefield. Believe in yourself that you can do the
activities given in this module.
Enclosed here is the step-by-step process on learning how to graph
polynomial functions. Be mindful of your answers and scores on the Pre-test
and compare it to your Post-test for this will measure your understanding level
on the lesson. You need to make it sure that you understand the flow of the
lesson so that you can have a better feedback later on.
As you begin, make it a commitment to yourself to answer each item
completely. This is for you and whatever you have learned will complete the
success of this module.
What I Need to Know
Welcome to another learning experience! As you go through this
module, you should be able to:
 identify the leading term, its leading coefficient and degree of
polynomial function
 determine the end-behavior of the graph of polynomial function using
the Leading Coefficient Test
 find the x and y intercepts of the function with the degree greater than 2
 determine the turning points of the graph
 describe the multiplicity of roots on the graph
 sketch the graph of polynomial functions
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
 Take your time reading the lessons carefully.
 Follow the directions and/or instructions in the activities
and exercises diligently.
 Answer all the given tests and exercises.
Icons of this Module
This module has the following parts and corresponding icons:
What I Need to
Know
This will give you an idea of the skills or
competencies you are expected to learn
in the module.
What I Know
This part includes an activity that aims to
check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip
this module.
What’s In
This is a brief drill or review to help you
link the current lesson with the previous
one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such as
a story, a song, a poem, a problem
opener, an activity or a situation.
What is It
This section provides a brief discussion of
the lesson. This aims to help you discover
and understand new concepts and skills.
What’s More
This comprises activities for independent
practice to solidify your understanding
and skills of the topic. You may check the
answers to the exercises using the
Answer Key at the end of the module.
What I Have
Learned
This includes questions or blank
sentence/paragraph to be filled into
process what you learned from the
lesson.
What I Can Do
This section provides an activity which will
help you transfer your new knowledge or
skill into real life situations or concerns.
Assessment
This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities
In this portion, another activity will be
given to you to enrich your knowledge or
skill of the lesson learned. This also tends
retention of learned concepts.
Answer Key
This contains answers to all activities in
the module.
At the end of this module you will also find:
References
This is a list of all sources used in
developing this module.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the
exercises.
2. Don’t forget to answer What I Know before moving on to the other
activities included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with
it.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in
mind that you are not alone.
We hope that through this material, you will experience meaningful
learning and gain deep understanding of the relevant competencies.
You can do it!
PRE TEST
Let us find out first what you have already probably known
related to the content of this module. Choose to answer all the
items.
Select the letter of your choice. Take note of your wrong answers and
find out what went wrong why you got such wrong answer. However, you may
skip this lesson if you get a perfect score without any help from others.
1. To define a polynomial function, what should be n if f(x) = xn?
A. any real number
C. a negative integer
B. a positive integer
D. any integer
2. Give the leading coefficient of the polynomial function y = x3 – 4x2 – 2 ?
A. 1
B. 2
C. 3
D. 4
3. The following are example of polynomial functions, EXCEPT?
A. f(x) = 3x5 + x3 + 2x +4
C. f(x) = 2x-4 – 4
B. f(x) = 3x2 +4x +2
D. f(x) = x3
4. Which polynomial functions in factored form shows the sketch of the graph
below?
A.
B.
C.
D.
5. Which of the following is the standard form of the polynomial function
f(x) =-10x2 + 9 + x4?
A. f(x) = 9 + x4-10x2
C. f(x) = 10x2 + x4 +9
B. f(x) = x4 + 9 +10x2
D. f(x) = x4 -10x2 + 9
6. Which of the choices below show the graph of polynomial function
y = x3 – 7x + 6?
A.
B.
c.
D.
7. Which of the following represents a graph of a polynomial function?
A.
B.
C.
D.
8. What is the factored form of the polynomial function f(x)= x 3 – 6x2 + 11x – 6
A. f(x) = (x-1)(x-2)(x-3)
B. f(x) = (x+1)(x+2)(x-3)
C. f(x) = (x+1)(x-2)(x-3)
D. f(x) = (x+1)(x+2)(x+3)
9. If the end behavior of a graph of the polynomial function falls both to the
left and to the right, which of the following is true about the leading term?
A. the leading coefficient is positive, the degree is odd
B. the leading coefficient is positive, the degree is even
C. the leading coefficient is negative, the degree is odd
D. the leading coefficient is negative, the degree is even
10. What is the y-intercept of the function y = x3 – 4x2 + x + 6
A. 2
B. -3
C. 6
D. -6
11. Determine the end behavior of the polynomial function f(x) = x5 – 3x4 + 4 ?
A. rises to the left, falls to the right
B. rises to the right, falls to the left
C. falls to both sides
D. rises to both sides
12. If you are to illustrate the graph of the polynomial function y= -3x4 – 6x +
4, which will be your possible sketch?
A.
B.
C.
D.
13. A point where the function changes from decreasing to increasing or from
increasing to decreasing values.
A. Multiplicity of a Root
C. Turning Points
B. Intercept of a Graph
D. Leading Coefficient Test
14. How will you sketch the graph of y = x (x-1)3 with respect to the x-axis?
A. tangent at both (1,0) and (0,0)
B. crossing (1,0) and tangent to (0,0)
C. tangent at (1,0) and crossing (0,0)
D. crossing both (1,0) and (0,0)
15. The following describes the graph of polynomial functions EXCEPT;
A. smooth
B. continuous
C. rounded turns
D. observable gaps
Lesson
1
THE LEADING COEFFICIENT
TEST
What I Need to Know
Welcome to another exciting topic on polynomial functions! This
lesson is a pre-requisite to the next lesson which is the graphing of polynomial
functions. In this lesson you are expected to:
 identify the leading term, its leading coefficient and the degree of
polynomial functions.
 determine the end behavior of the graph of polynomial functions
using the Leading Coefficient Test.
What I Know
This activity is to assess how much you have already known
about the lesson. Choose the letter of your answer. Let your facilitator check
your answers and take note of the missed items. However, you may skip this
lesson if you get a perfect score without any help from others.
1. What is the leading term of the polynomial function y = -4x3 + x4 – 2x -3?
A. -4x3
B. x4
C. 2x
D. -3
2. Which polynomial functions in factored form shows the sketch of the graph
below?
A.
B.
C.
D.
1
3. Which of the following is true about the leading coefficient and degree of y
= x4 - 4x3 – 2x -3?
A. positive, odd
B. negative, even.
C. positive, even.
D. negative, odd
4. Which of the following graph represents a polynomial function?
A.
B.
C.
D.
5. Determine the end behavior of the graph of polynomial function
f(x) = -x5 – 3x4 + 4?
A. rises to the left, falls to the right B. rises to the right, falls to the left
C. falls to both sides
D. rises to both sides
6. What is the y-intercept of the function f(x) =
A. -18
B. -9
C. -1
D. 9
?
7.. What is the degree of the polynomial function y =
A. 0
B. 1
C. 2
D. 3
?
8. What is the y- intercept of the function y = x3 – 4x2 + 3x – 2?
A. -4
B. -2
C. 2
D. 4
9. The following are example of polynomial functions, EXCEPT?
A. f(x) = 3x5 + x3 + 2x +4
B. f(x) = 3x2 +4x +2
C. f(x) = 2x4 – 4
D. f(x) = x3 + 2/(x+4)
10. Which of the following graph could be the illustration of y= -3x4 – 6x + 4?
A.
B.
C.
2
D.
11. If the end behavior of a graph of the polynomial function rises to the left
and falls to the right, which of the following is true about the leading term?
A. the leading coefficient is positive, the degree is odd
B. the leading coefficient is positive, the degree is even
C. the leading coefficient is negative, the degree is odd
D. the leading coefficient is negative, the degree is even
12. To define a polynomial function, what should be n if f(x) = xn?
A. a positive integer
B. any real number
C. a negative integer
D. any integer
13. Which of the following is true about the graph below?
A. the leading coefficient is positive, the degree is odd
B. the leading coefficient is positive, the degree is even
C. the leading coefficient is negative, the degree is odd
D. the leading coefficient is negative, the degree is even
14. If the leading coefficient is positive and the degree is even, which would
be its possible graph?
A.
B.
C.
D.
15. What is the degree of the polynomial function y = -4x3 + x4 – 2x -3?
A. -4
B. -3
C. 3
D. 4
3
What’s In
In order to get you ready in this lesson, do the next activity. It is
about the concepts of polynomial function which you have
learned in the previous module.
Leading
Term
Polynomial Function
Leading
Coefficient
Degree
y = 3x4 – 8x3 + 8x – 6
1.
2.
3.
y = -x4 + 16x + 2
4.
5.
6.
y = x3 – 4x2 + x + 6
7.
8.
9.
y = -2x5 + x3 – 7x2 + 4
10.
11.
12.
What’s New
Let’s explore!
Observe the graphs given below and answer what is asked for
each item. The first one is done for you. One point check is
given in each blank.
Leading term:
anxn where an ≠ 0
Leading coefficient : an ˃ 0  positive
an < 0  negative
Degree :
even or odd
A. Polynomial function: y = x3 – 7x + 6
X
Leading term: ___________
1
Leading coefficient : __(Positive/Negative)
3
Degree :___(Even/Odd)
Left: (Rise/Fall)
Right: (Rise/Fall)
2
Number of turn point: _____
3
4
B. Polynomial function: y= -3x3- 2x2 + 8x
Leading term: ___________
Leading coefficient : ______________
Degree :______________
C. Polynomial functions: y = x4 – 3x2 – 3
Leading term: ___________
Leading coefficient : ______________
Degree :______________
D. Polynomial Functions: y = -3x2 – 6x + 4
Leading term: ___________
Leading coefficient : ______________
Degree :______________
E. Polynomial functions: f(x) = x3 + 4
Leading term: ___________
Leading coefficient : ______________
Degree :______________
Were you able to provide what is asked for each blank? Were you able
5
to recall the concept of polynomial function? Now that you have
reviewed the important concepts of polynomial function, let us explore
some methods on showing the graph of a polynomial function as you go
What Is It
The activity that you have just answered will lead you to the new
topic which is The Leading Coefficient Test. This test can help you
determine the end behavior of the graph of polynomial functions by looking at
the degree and the leading coefficient so that it will give you a rough sketch of
the graph. Let’s start by reviewing the definition of a polynomial function that
you have learned in the previous topic.
A polynomial function is a function of the form :
F(x) = an xn + an-1 xn-1 + … + a1 x + a0
where an are real numbers called coefficients and n is a positive
integer, and anxn is the leading term, an is the leading coefficient, a0 is the
constant term and an ≠ 0.
End behavior is a description of the values of the function as x
approaches positive infinity or negative infinity. The leading coefficient and the
degree of a polynomial function determine its end behavior and it is very much
helpful in graphing polynomial function.
Polynomial End Behavior
Leading
Coefficient
Positive
a>0
Degree
Graph
Comparison
Even
The graph rises
to the right and
rises to the left
Negative
a<0
Even
The graph falls
to the right and
falls to the left
6
End Behavior
Positive
a>0
Odd
The graph rises
to the right and
falls to the left
Negative
a<0
Odd
The graph rises
to the left and
falls to the right.
Here are the examples of the four cases of the Leading Coefficient
Test.
A. On the positive leading coefficient and even degree.
Steps
Expression
1. Look at the given
polynomial function
f(x)= 2x4 – 3x3 + x -1
2. Identify the leading
term, its leading
coefficient and degree
2x4
3. Identify the end
behavior of the graph.
2x4
Discussion
2 is the leading
coefficient and it is
positive.
4 is the degree and it is
an even number.
The graph rises to the
right and rises to the
left
4. Graph
The leading coefficient
of the polynomial is
positive and its degree
is an even number.
B. On the negative leading coefficient and even degree.
Steps
Expression
7
Discussion
1. Look at the given
polynomial function
f(x)= -3x2 – 6x + 4
2. Identify the leading
term, its leading
coefficient and degree
-3x2
3. Identify the end
behavior of the graph.
4.Graph
-3x2
-3 is the leading
coefficient and it is
negative.
2 is the degree and it is
an even number.
The graph falls to the
right and falls to the left
The leading coefficient
of the polynomial is
negative, and its degree
is an even number.
C. On the positive leading coefficient and odd degree.
Steps
Expression
1. Look at the given
polynomial function
f(x)= x3 + 5
2. Identify the leading
term, its leading
coefficient and degree
x3
3. Identify the end
behavior of the graph.
4. Graph
x3
Discussion
1 is the leading coefficient
and it is positive.
3 is the degree and it is
an odd number.
The graph rises to the
right and falls to the left
The leading coefficient of
the polynomial is positive
and its degree is an odd
number.
D. On the negative leading coefficient and odd degree.
Steps
Expression
Discussion
8
1. Look at the given
polynomial function
f(x)= -x5 + x4 -2x3 + 1
2. Identify the leading
term, its leading
coefficient and degree
-x5
3. Identify the end
behavior of the graph.
-x5
-1 is the leading
coefficient and it is
negative.
5 is the degree and it is
an odd number.
The graph rises to the
left and graph falls to
the right.
4.Graph
The leading coefficient
of the polynomial is
negative and its degree
is an odd number.
Were you able to understand the concept of the Leading Coefficient Test? Do
you want more examples? Try to do the next activity.
What’s More
Let’s try this!
Determine the end behavior of the graph of each polynomial function.
Show your answer by filling in the blanks and show the possible sketch of the
graph. The first one is made for you. One point check is given in each blank.
Sketching the graph would earn 3 points.
GUIDED PRACTICE
A. y = -x4 + 10x2 - 9
1. The leading term is -x4
2. The leading coefficient is -1 and it is negative.
3. The degree is 4 and it is even.
4. Since the leading coefficient is negative and the degree is even, then
the graph falls to the left and falls to the right.
5. Possible sketch:
9
B. y = -2x3 + x2 + 18x – 9
1. The leading term is ____
2. The leading coefficient is ______ and it is _________.
3. The degree is _____ and it is ______.
then
4. Since the leading coefficient is _________ and the degree is ____,
the
graph _______ to the left and _______ to the right.
5. Possible sketch:
C. y = 4x3 – 4x2 – 19x +10
1. The leading term is ____
2. The leading coefficient is ______ and it is _________.
3. The degree is _____ and it is ______.
4. Since the leading coefficient is _________ and the degree is ____,
then the
graph _______ to the left and _______ to the right.
5. Possible sketch:
D. y = x4 – 10x2 + 9
1. The leading term is ____
2. The leading coefficient is ______ and it is _________.
3. The degree is _____ and it is ______.
4. Since the leading coefficient is _________ and the degree is ____,
then the
graph _______ to the left and _______ to the right.
5. Possible sketch:
10
Have you answered the guided practice given above? If yes, very
good! How was your score? So here are the remarks as to the number
of scores that you got in the activity that you have just answered.
If your score is:
36: You got the perfect score!
25-35 : You are doing great! Review your missed items and proceed to
the next activity.
15-24: You are doing just fine but you need to gain mastery on the
Leading Coefficient Test.
1-14 : Please consider study harder. Ask help form your friend or your
teacher or even a relative to make you understand the concepts.
Revisit the given examples and answer again.
INDEPENDENT PRACTICE
A. Identify whether the function graphed has positive or negative leading
coefficient
1.
2.
3.
4.
B. Determine whether the function graphed has an odd or even degree
5.
6.
7.
8.
11
What I Have Learned
Now, in order to gain more mastery, complete the table below.
Polynomial Functions
Leading Coefficient
(positive/negative)
Degree
(odd/
even)
End behavior of
the Graph
Left
side
f(x) = 3x3 – 8x2 -8x +8
f(x) = -x4 - 6x3+9x2-6x+8
f(x) = -x3 – 4x2 + x+ 6
1.
2.
3.
4.
5.
6.
7.
8.
9.
Possible
Sketch
Right
side
10.
11.
12.
13.
14.
15.
What I Can Do
It’s time for you to check your understanding. Act like a detective
and do the activity below. Read the situation and answer the
questions that follow.
Two students namely Amihan and Pirena were asked by their teacher to
show the sketch of the graph of the polynomial function f(x) = x4 – 3x2 -3.
Who among the two students show the graph correctly? Why do you say
so? Explain your answer.
PIRENA
AMIHAN
12
Were you able to identify who between the two students graph it
right? If yes, you did a great job! If no, try harder next time.
Assessment
If you wish to gain more mastery, try these activities below.
GUIDED PRACTICE
Identify the leading coefficient, degree and end-behavior.
y = -4x4 – 3x3 + x2 +4
1. leading coefficient:
2. degree:
3. end behavior:
y = 3x2 + 6x – 10
4. leading coefficient:
5. degree:
6. end behavior:
7
y= -2x + 6x5 + 2x3
7. leading coefficient
8. degree:
9. end behavior
5
y = x – 4x2 + 3x – 1
10. leading coefficient:
11. degree:
12. end behavior:
y = 5.5x8 + 7.5x4
13. leading coefficient:
14. degree:
15. end behavior:
INDEPENDENT PRACTICE
13
Identify whether the function graphed has an odd or even degree and a
positive or negative leading coefficient.
1.
5.
9.
13.
2.
3.
6.
10.
14.
4.
7.
8.
11.
12.
15.
14
Additional Activities
For your mastery on the Leading Coefficient Test, do the next
activity below.
Write the polynomial function indicated by the description in each item.
Choose your answer from the box.
f(x) = 5x3 + 9x + 1
f(x) = 2x6 + 3x4 + 5x2
f(x) = -4x2 + 3x -1
f(x) = -5x3 + x - 8
__________1. The end behavior of the graph of this polynomial function rises
to both left and right.
__________2. The end behavior of the graph of this polynomial function rises
to the left and falls to the right.
__________3. The leading coefficient is negative, and the degree is even.
__________4. The leading coefficient is positive, and the degree is even.
__________5. The end behavior of the graph of this polynomial function rises
to the right and falls to the left.
__________6. The leading coefficient is positive, and the degree is odd.
__________7. The end behavior of the graph of this polynomial function falls
to both left and right.
__________8. The leading coefficient is negative, and the degree is odd.
f(x) = 2x3 + 3x2 – 4x
f(x) = -x5 + 5x4 + 1
f(x) = 3x6 + 4x5 - 2x + 4
f(x) = -3x4 + 8x2 + 5
15
__________9. The end behavior of the graph of this polynomial function rises
to both left and right.
__________10. The end behavior of the graph of this polynomial function
rises to the left and falls to the right.
__________11. The leading coefficient is negative, and the degree is even.
__________12. The leading coefficient is positive, and the degree is even.
__________13. The end behavior of the graph of this polynomial function
rises to the right and falls to the left.
__________14. The leading coefficient is positive, and the degree is odd.
__________15. The end behavior of the graph of this polynomial function falls
to both left and right.
__________16. The leading coefficient is negative, and the degree is odd.
16
Lesson
2
GRAPHING
POLYNOMIAL
FUNCTIONS
What I Need to Know
This is it! You are now about to learn more on polynomial
functions because as you go through this lesson, you should be able to:
 find the x and y intercepts of a polynomial function with
degree greater than 2
 determine the turning points of the graph
 describe the multiplicity of roots on the graph
 sketch the graph of polynomial functions
What I Know
Earlier in the study of your Grade 10 Mathematics, you learn about
polynomials, the Factor Theorem, the Rational Root Theorem, the Remainder
Theorem and the Fundamental Theorem of Algebra.
This time answer the activity below for you to know how much you have
already learned about the lesson. However, if you get a perfect score without
17
any help from others, you may skip this lesson. You can continue working all
the activities in this lesson at your own desire.
1. Which of the following is the polynomial function given if the factored form
is y = x(x+4)(x+1)(x-3)?
A. y = x4 + 2x3 – 11x2 – 12x
B. y = 2x3 – 11x2 – 12x
C. y = 11x2 – 12x
D. y = 11x3 – 12x
2. Which function represents the graph below?
A. f(x) = x(x+4)(x+1)(x-3)
B. f(x) = (x+4)(x+1)(x-3)
C. f(x) = x(x+4)(x+2)
D. f(x) = x(x+4)(x+2)(x-1)
3. Which of the following could be the graph of y = x3 – 3x2 – x + 3?
A.
B.
C.
D.
4. What is the factored form of the polynomial function y = x3 – 3x2 – x + 3?
A. y = x(x-2)(x+2)
B. y = (x+1)(x-3)(x-1)
C. y = (x+1)(x-3)(x+1)
D. y = (x+1)(x-3)(x-3)
5. What is the y – intercept of the function f(x) = x(x+4)(x+1)(x-3)?
A. -4
B. -3
C. -1
D. 0
6. How many turning points does a cubic function with 3 real zeros have?
A. -1
B. 0
C. 1
D. 2
7. Which polynomial function has zeros -1, 3, 1?
A. y = (x+1)(x-3)(x+1)
B. y = (x+1)(x-3)(x-3)
C. y = (x+1)(x-3)(x-1)
D. y = (x-1)(x-3)(x-1)
8. Which polynomial function has a y-intercept of 8?
18
A. f(x) = x3 + x2 – 10x + 8
B. f(x) = x3 + x2 – 10x – 8
C. f(x) = x3 + x2 – 10x – 4
D. f(x) = x3 + x2 – 10x + 4
9. What is the multiplicity of the root -1 in the equation y = x4-2x3-3x2+4x+4?
A. 1
B. 2
C. 3
D. 4
10. What is the degree of the polynomial function y = 2x3-3x2+4x+4?
A. 1
B. 2
C. 3
D. 4
11. What are the x-intercepts of the function y = x4 + x3 – 7x2 – x + 6?
A. 1, -1, -3, 2
B. 1, -1, 2
C. 1, -1, 3
D. 1, -3, 2
12. Which of the following shows the graph of y = x(x+2)(x+1)?
A.
B.
C.
D.
13. If the end behavior of a graph of the polynomial function rises to the right
and falls to the left, which of the following is true about the leading coefficient
and its degree?
A. negative, even
B. positive, even
C. negative, odd
D. positive, odd
14. How will you sketch the graph of y = x (x-1)4 with respect to the x-axis?
A. tangent at both (1,0) and (0,0)
B. crossing (1,0) and tangent to (0,0)
C. tangent at (1,0) and crossing (0,0)
D. crossing both (1,0) and (0,0)
15. What is the leading coefficient of the function y = x4-2x3-3x2+4x+4?
A. 1
B. 2
C. 3
D. 4
What’s In
Here is another activity to hone your knowledge in finding
the roots of polynomial functions. Find the factors of the given
19
polynomial function and write its roots on the last column. The first
one is done for you.
Polynomial Function
1. y = x4 + x3 - 7x2 – x + 6
Factors
Roots
y=(x+3)(x-2)(x+1)(x-1)
-3,2,-1,1
2. y = x3 + 9x2 + 15x – 25
3. y = x4 + 4x3 – 13x2 -40x + 48
4. y = x4 + 6x3 – 9x2 - 54x
5. y = x4 – 9x3 –x2 + 9x
How was it? It’s quiet challenging right? But of course if you have
mastered synthetic division or the rational root theorem, it’s just a piece of
cake! So let’s take a look at your score. if your score is:
8 : You are doing great! You have mastered the previous theorems
5-7: You are doing just fine but you need to practice more in order to gain
mastery.
1-4 : Please consider study harder. Ask help form your friend or your
teacher or even a relative to make you understand the concepts. Revisit
the given examples and answer again.
What’s New
Using your graphing paper or graphing notebook, try to graph
the function given below by plotting the values of x and y
presented by the table.
1. f(x) = x2
X
-3
-2
-1
0
1
2
3
y
9
4
1
0
1
4
9
2.f(x) = x3
X
-3
-2
-1
0
1
2
3
y
-27
-8
-1
0
1
8
27
20
Did you show the graph correctly? You were taught how to graph in
your previous years of studying Mathematics. This lesson will help you
deepen your understanding in graphing. This time it’s more of a “level up”
stage since you will be graphing polynomial functions with a degree greater
than 2. Let’s get started.
What Is It
In this lesson, you will be provided with illustrative examples and
the step- by- step procedure on graphing polynomial functions.
Here are some important details that must be kept in mind while
graphing polynomial functions. Use graphing paper or notebook when plotting.
1. The graph of a polynomial function is continuous because every
polynomial function with real coefficients has the set of real
numbers as its domain.
Furthermore, it means that there are no holes or gaps in which you
can even draw the graph without lifting your pen. Also, the graph of
polynomial function is smooth, it has no sharp corner and the turns
are rounded.
2. Every polynomial function of the nth degree cannot have more than
n roots (Fundamental Theorem of Algebra), which means the graph
cannot intersect the x-axis more than n times.
3. A polynomial function of degree n has n-1 turning points on its
graph or the number of turning points is always less than n.
4. The multiplicity of root r is the number of times that x-r is a factor of
f(x). When a real root has even multiplicity, the graph of f(x) is
tangent to the x-axis. When a real root has odd multiplicity greater
than 1, the graph bends as it crosses the x-axis.
Example 1: Sketch the graph of f(x) = x3 – 2x2 – x + 2
Steps
1. Determine the
end behavior of the
graph
2. Find the x
Expression
f(x) =
x3
–
2x2
Discussion
–x+2
Let f(x)= 0
i. f(x)= x3 – 2x2 – x + 2
21
The leading coefficient is 1
and it is positive, and the
degree is 3 and it is an odd
number, thus the graph rises
to the right and falls to the
left.
Equate the equation to zero.
intercept
3. Find the y
intercept
0 = x3 – 2x2 – x + 2
0= (x+1)(x-1)(x-2)
x+1=0; x-1=0 & x-2=0
x= -1 x= 1
x=2
(-1,0)
(1,0) (2,0)
Apply the Zero Product
Property to get the xintercepts.
Therefore , the graph will
pass through (-1,0)
(1,0)
(2,0).
Let x= 0
f(x) = x3 – 2x2 – x + 2
f(x)= (0)3 – 2(0)2 – 0 + 2
f(x)= 0 –0 – 0 + 2
f(x)= 2
(0,2)
Substitute x to zero (0) and
simplify
4. Sketch the graph
The graph will also pass
through (0, 2).
You may also find other
points on the graph by
making a table of values.
X -3
-2
-1
0
1
2
3
40
-12
0
2
0
0
8
y
Notice that your graph of f(x) = x3 – 2x2 – x + 2 crosses the x- axis 3
times, it’s because the degree of your function is 3. A polynomial function of
the nth degree cannot have more than n roots. This means that the graph
cannot intersect the x-axis more than n times. It is true that the graph falls to
the left and rises to the right as we have mentioned in Step 1 using the
Leading Coefficient Test.
The turning points of a graph occur when the function is changing
values, from decreasing to increasing or from increasing to decreasing. A
polynomial function of degree n has n-1 turning points on its graph or the
number of turning points is always less than the degree n. The graph above
illustrates 2 turning points since the degree of f(x) is 3.
Sometimes a polynomial functions has a factor that occurs more
than once. This makes a multiple root. The multiplicity of root r is the number
of times that x-r is a factor of f(x). When a real root has even multiplicity, the
graph of f(x) is tangent to the x-axis. When a real root has odd multiplicity
greater than 1, the graph bends as it crosses the x-axis. Consider the next
example.
Example 2:Sketch the graph of y = (x+1)(x+2)2(x-1)3
22
Steps
1. Determine the
end behavior of the
graph.
If you get the
product of the
polynomials, the
leading term will be
x6.
2. Find the x
intercept.
(The given is in
factored form
already, there’s no
need to factor the
polynomial
completely.)
Expression
y=
Discussion
The leading coefficient is
1 and it is positive, the
degree is 6 and it is an
even number, thus the
graph rises to the right
and rises to the left.
(x+1)(x+2)2(x-1)3
Let y= 0
i. y= (x+1)(x+2)2(x-1)3
0 = (x+1)(x+2)2(x-1)3
(x+1)=0 (x+2)2=0
(x-1)3=0
x+1=0;
x= -1
(x+2)2=0
Equate the equation to
zero
Applying the Zero
Product Property.
=
x- 1= 0
x=1
Equate each factor to
zero and simplify
=
x+2=0
x = -2
(-1,0), (-2,0), (1,0)
3. Find the y
intercept
Let x= 0
y= (x+1)(x+2)2(x-1)3
y= (0+1)(0+2)2(0-1)3
y= (1)(2)2(-1)3
y=-4
(0,-4)
4. Sketch the graph
23
The graph will pass
through (-1,0), (-2,0),
(1,0)
Equation
Substitute x to zero (0)
and simplify
The graph will also pas
through (0,-4)
You may also find other
points on the graph by
making a table of values.
X -2 -1 0 1 2
y 0 0 -4 0 48
Notice on the graph where your zero is -2, the given factor is (x+2)2,
meaning to say the factor (x+2) occurred twice. When a real root has even
multiplicity, the graph of the function is tangent the x-axis, it simply touches it
but does not cross the x-axis. This is true for even multiplicity (the exponent is
even). Thus -2 is of even multiplicity 2. Observe the graph where the zero is 1
where the given factor is (x-1)3. When a real root has odd multiplicity, the
graph bends as it crosses the x-axis, hence 1 is an odd multiplicity 3.
Look at this table for more illustration.
Roots
Multiplicity
-2
Even
Polynomial functions Leading
Term
-1
1
End
behavior
of the
graph
Odd
Odd
Behavior of the graph relative
to x-axis
Is tangent to
factors
Roots
yinterce
pts
Turning
points
Crosses
Crosses
You are now equipped with knowledge on how to sketch the graph of
polynomial functions. It’s now your turn to do the next activity.
What’s More
Here is an activity to help you check your understanding on the
lesson. Complete the table below.
GUIDED PRACTICE
You start with the column with the given so that it would be easy for
you to fill out the other parts of the table.
24
y=x4-2x3 -7x2+8x+12
1.
2.
3.
4.
5.
6.
7.
8.
9.
(x-2) (x+2)
(x-3)(x+3)
10.
11.
12.
13.
14.
15.
16.
-2,1,5
17.
18.
INDEPENDENT PRACTICE
Sketch the graph of the given polynomial functions.
1. y = x4-2x3 -7x2+8x+12
2. y = (x-2) (x+2) (x-3)(x+3
3. y = (x+2)(x-1)(x-5)
What I Have Learned
In the previous module, you have learned that one of the ways
to get the factor of polynomial function is through Division. But what is the
mathematical name for the division symbol (÷)? To find put, do the activity
below.
Match the letter to its corresponding answer and write the letter below.
Column A
Column B
B: a given factor of polynomials with a
root of even multiplicity
A. y = x3 + 7x2 + 7x – 15
L: a given factor of polynomials with a
root of odd multiplicity
B. y = (x-2)2(x-1)4
S: polynomial function with 3 turning
points
C. y = -x5 + 3x4+x3- 7x2+4
O: polynomial function with (x+3) as
one of the factor
D. f(x) = (x+1)(x-1)3
E. y= x3 + 8x2 +17x +10
E: This graph’s end behavior rises to
the left but falls to the right.
F. f(x) = -x4+2x3+13x2-14x-
U: polynomial function with -2 as one
of its roots.
24
25 ____
____ ____ ____ ____ ____
A
B
C
D
E
F
Thus, the mathematical name of the division symbol is ___________.
Well, the activity that you have just answered contains some important
details in graphing polynomial functions. It is important that you keep them in
your mind so that you can have a broader idea and a bigger picture in your
mind on the sketch of your graph
What I Can Do
Now it’s your turn. For each of the given polynomial function, do
the following and sketch the graph.
1.
2.
3.
4.
5.
describe the leading term (2 pts)
determine the end behaviors (2 pts)
find the x-intercepts and y-intercepts (3 pts)
create table of values (3 pts)
sketch (4 pts)
i. y = x4 + 5x3 + 2x2 – 8x
ii. y = (x+2) ( 2x+1) (x+3)
Have you sketch the graph of the polynomial functions given
above? I hope you enjoy doing the activities. The next activities will bring you
more items to test your mastery, so keep going!
Assessment
GUIDED PRACTICE
Complete the table below.
Polynomial Functions
Roots
Sketch
26
Turning
Points
1. . y = (x-1)(x+1)(x-3)(x-2)
2. y = x3 + 7x2 + 7x – 15
3. y = (x-3)(x-1)(2x-1)
INDEPENDENT PRACTICE
It’s time to do your own thing. Examine the graph below and answer
the question given.
Give at least 2 out of 3 different reasons why the graph below cannot be the
graph of f(x) = x4 + x2 + 1.
27
Additional Activities
Below are set of activities that will help you gain mastery in
finding the coordinates on the graph.
Sketch the graph of the following polynomial functions with a credit
of 3 points each graph.
11. f(x) = x4 + x3 – 5x2 – x +6
12. f(x) = x5 + 2x4 – x3 – 6x2 – 4x
13. f(x) = 2x3 - 5x + 7
14. f(x) = x3 + 8x2 + 20x + 16
15. f(x) = x3 - 5x2 + 6x – 5
Summary
 The Leading Coefficient Test is a test to determine the end behavior of
the graph of polynomial function by describing the leading term. It
comes with 4 cases.
1. If the degree of the polynomial is odd and the leading coefficient
is positive, then the graph falls to the left and rises to the right.
2. If the degree of the polynomial is odd and the leading coefficient
is negative, then the graph rises to the left and falls to the right.
3. If the degree of the polynomial is even and the leading
coefficient is positive, then the graph rises to the left and rises to
the right.
4. If the degree of the polynomial is even and the leading
coefficient is negative, then the graph falls to the left and falls to
the right.
 The graph of a polynomial function is continuous and smooth and has
rounded turns.
 A polynomial function of degree n has n-1 turning points.
28
 Every polynomial function of the nth degree cannot have more than n
roots.
 Here are the steps to sketch the graph of a polynomial function
1. Identify the end behavior of the graph through the Leading
Coefficient Test.
2. Factor the polynomials completely (if the given is not in factored
form)
3. Find the x- intercepts as well as the y-intercepts to determine
where on the x and y axis the graph passes through.
4. Plot the other points.
 A zero has a "multiplicity", which refers to the number of times that its
corresponding factor appears in the polynomial. When a real root has
even multiplicity, the graph of f(x) is tangent to the x-axis. When a real
root has odd multiplicity greater than 1, the graph bends as it crosses
the x-axis.
29
POST-TEST
Choose the letter of the correct answer.
1. Which of the following graphed function has the negative leading coefficient
and an even degree?
A
B
C
D
2. What is the factored form of the polynomial function
y = x4 + x3 – 7x2 – x + 6 ?
A. f(x) = (x-1)(x-2)(x-3)(x-2)
C. f(x) = (x+1)(x-2)(x-3)(x+3)
B. f(x) = (x+1)(x+2)(x-3)(x-3)
D. f(x) = (x+1)(x-2)(x+3)(x-1)
3. If the end behavior of a graph of the polynomial function rises both to the
left and to the right, which of the following is true about the leading term and
degree?
A. the leading coefficient is positive, the degree is odd
B. the leading coefficient is positive, the degree is even
C. the leading coefficient is negative, the degree is odd
D. the leading coefficient is negative, the degree is even
4. What is the y- intercept of the function y = (x+2)(x-2)(x-4) ?
A. -16
B. 8
C. 16
D. 32
5. How will you sketch the graph of y = x (x-1)4 with respect to the x-axis?
A. sketch it tangent at both (1,0) and (0,0)
B. sketch it crossing (1,0) and tangent to (0,0)
C. sketch it tangent at (1,0) and crossing (0,0)
D. sketch it crossing both (1,0) and (0,0)
6. Give the leading coefficient of the polynomial function
y = – 4x2+3x3+ 3x – 2 ?
30
A. -4
B. 2
C. 3
D. 4
7. Which of the following is the graph of y = x3 – 5x2 + 6x – 5?
A.
B.
C.
D.
8. Which of the following represents a polynomial function?
A. y =
B. y = 4/(x+2) +1
C. y = 4x2.5+8x-3+2x+3
D. y = 3x3 – 7x2 +4x – 2
9. Which function could describe the graph?
A. f(x) = 2x5 – x2 + 75
B. f(x) = -5x3 + x – 8
C. f(x) = x2 + 5x + 6
D. f(x) = -x4 + 3x3 – 4x + 2
10. How many turning points will a quartic function with four real zeros have?
A. 1
B. 2
C. 3
D. 4
11. What is the multiplicity of the root 2 in the equation y = x4-2x3-3x2+4x+4?
A. 1
B. 2
C. 3
D. 4
12.Which polynomial function has zeros -2, ½, 2, 1?
A. y = 2x4 – 3x3 – 7x2 +12x – 4 B. y = 2x3 – 3x2 – 7x +12
C. y = 3x3 – 7x2 +12x – 4
D. y = 3x4 – 7x2 +12x – 4
13. To define a polynomial function, what should be n if f(x) = xn?
A. any real number
C. a positive integer
B. a negative integer
D. any integer
14. Determine whether the given binomial is a factor of the polynomial f(x).
A. (x-3); f(x)= x3+2x-3
C. (x+4); f(x) = 2x4+8x3+2x+8
B. (x-2); f(x) = x4-2x3+3x2+4x+4
D. (x+5); f(x) = x4-2x3+3x2+4x+25
15. What is the degree of the polynomial function y = x3 – 4x2 + 3x – 2 ?
A. 1
B. 2
C. 3
D. 4
31
Answer Key
PRETEST
1. B
2. A
3. C
4. A
5. D
6.A
7.C
8.A
9.D
10.C
11.A
12.A
13.C
14.D
15.D
LESSON 1
WHAT I KNOW
1.A
2. B.
3. C
4. D
5. A
6. B
7. D
8. B
9. D
10. A
11. C
12. A
13. C
14. C
15.D
WHAT’S IN
1. 3x4
2. 3
3. 4
4. –x4
5. -1
6. 4
7. x3
8. 1
9. 3
10. -2x5
11. -2
12. 5
WHAT’S NEW
B. Polynomial function: y= -3x3- 2x2 + 8x
Leading term: -3x3
Leading coefficient : -3, negative
Degree : -3, odd
C. Polynomial functions: y = x4 – 3x2 – 3
Leading term: x4
Leading coefficient : 1, positive
Degree : 4, even
D. Polynomial Functions: y = -3x2 – 6x + 4
Leading term: -3x2
Leading coefficient : -3, negative
Degree : 2, even
32
E. Polynomial functions: f(x) = x3 + 4
Leading term: x3
Leading coefficient : 1, positive
Degree : 3, odd
WHAT’S MORE
GUIDED PRACTICE
B. y = -2x3 + x2 + 18x – 9
1. The leading term is -2x3
2. The leading coefficient is -2 and it is negative.
3. The degree is 3 and it is odd.
4. Since the leading coefficient is negative and the degree is od d, then the
graph rises to the left and falls to the right.
5. Possible sketch:
C. y = 4x3 – 4x2 – 19x +10
1. The leading term is 4x3
2. The leading coefficient is 4 and it is positive.
3. The degree is 3 and it is odd.
4. Since the leading coefficient is positive and the degree is odd, then the
graph falls to the left and rises_ to the right.
5. Possible sketch:
D. y = x4 – 10x2 + 9
1. The leading term is x4
2. The leading coefficient is 1 and it is positive.
3. The degree is 4 and it is even.
4. Since the leading coefficient is positive and the degree is even_, then the
graph rises to the left and rises to the right.
5. Possible sketch:
Possible Sketch B,C,D,
INDEPENDENT PRACTICE:
1.
2.
3.
4.
5.
6.
7.
8.
Negative
Negative
Positive
Negative
Odd
Odd
Even
Odd
33
WHAT I HAVE LEARNED
Polynomial Functions
f(x) = 3x3 – 8x2 -8x +8
f(x) = -x4 - 6x3+9x2-6x+8
f(x) = -x3 – 4x2 + x+ 6
Leading Coefficient
(positive/negative)
1. positive
2. negative
3. negative
Degree
(odd/even)
4. odd
5.even
6.odd
Behavior of the Graph
Left side
7.falling
8. falling
9. rising
Right side
10. rising
11. falling
12. falling
WHAT I CAN DO
Amihan shows the graph correctly because it follows the Leading Coefficient Test. From the
function, the leading coefficient is positive and the degree is even, therefore the end behavior of the
graph on the left side is rising as well as on the right side and Amihan shows the graph correctly.
ASSESSMENT:
GUDED PRACTICE
1. negative
2. Even
3. Falls to the left, falls to the right
4. positive
5. Even
6. Rises to the left , rises to the right
7. negative
8.odd
9. Rises to the left, falls to the right
10. positive
11. Odd
12. Falls to the left, rises to the right
13. positive
14. Even
15. Rises to the left, rises to the right
INDEPENDENT PRACTICE
1. negative leading coefficient – odd degree
2. Negative leading coefficient – even degree
3. positive leading coefficient – even degree
4. Negative leading coefficient- even degree
5. negative leading coefficient – even degree
6. Positive leading coefficient – odd degree
7. negative leading coefficient – even degree
8. Negative leading coefficient – odd degree
9. negative leading coefficient – even degree
10. Negative leading coefficient – odd degree
11. negative leading coefficient – odd degree
12. Negative leading coefficient – odd degree
13. positive leading coefficient – even degree
14. Positive leading coefficient – odd degree
15. positive leading coefficient – even degree
34
ADDITIONAL ACTIVITIES
1. f(x) = 2x6 + 3x4 + 5x2
2. f(x) = -5x3 + x – 8
3. f(x) = -4x2 + 3x – 1
4. f(x) = 2x6 + 3x4 + 5x2
5. f(x) = 5x3 + 9x + 1
6. f(x) = 5x3 + 9x + 1
7. f(x) = -4x2 + 3x – 1
8. f(x) = -5x3 + x – 8
9. f(x) = 3x6 + 4x5 - 2x + 4
10.f(x) = -x5 + 5x4 + 1
11. f(x) = -3x4 – 8x2 + 5
12. f(x) = 3x6 + 4x5 - 2x + 4
13. f(x) = 2x3 +3x2 – 4x
14. f(x) = 2x3 +3x2 – 4x
15.f(x) = 2x3 +3x2 – 4x
16. f(x) = -x5 + 5x4 + 1
LESSON 2
WHAT I KNOW
1. A
2. C
3. A
4. B
5. D
6. D
7. C
8. A
9. B
10. C
11. A
12. B
13. D
14. C
15. A
WHAT’S IN
Polynomial Function
Factors
Roots
7x2 –
y=(x+3)(x-2)(x+1)(x-1)
-3,2,-1,1
y = (x+5)(x-5)(x-1)
5, -5, 1
y= (x+4)(x+4)(x-3)(x-1)
-4 (2 times), 3,1
y= (x(x-3)(x+3)(x+6
0, 3, -3, -6
y = x(x-9)(x+1)(x-1)
0, 9, 1, -1
1. y =
x4 +
x3 -
2. y =
x3 +
9x2 +
3. y =
x4
4x3
+
–
x+6
15x – 25
13x2
-40x + 48
4. y = x4 + 6x3 – 9x2 - 54x
5. y =
x4
–
9x3
–x2
+ 9x
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WHAT’S NEW
Here is the graph of f(x) = x2 and f(x) =x3
WHAT’S MORE
GUIDED PRACTICE
y = x4 – 2x3 – 7x2 + 8x + 12
1. x4
2. rises to the left, rises to the right
3.(x+1)(x-3)(x-2)(x+2)
4. -1, 3, 2, -2
5. 12
6. 3
7. y = x4 – 13x2 + 36
8. x4
9. rises to the left, rises to the right
10. 2, -2, 3, -3
11. 36
12. 3
13. y = x3 – 4x2 – 7x + 10
14. x3
15. falls to the left, rises to the right
16. (x+2)(x-1)(x-5)
17. 10
18. 2
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INDEPENDENT PRACTICE
1. y = x4 + x3 – 7x2 –x + 6
1. y = (2x+3)(x-2)(x-1)
1. y = x3 – 4x2 – 7x + 10
WHAT I HAVE LEARNED
OBELUS – the mathematical name of the division symbol.
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WHAT I CAN DO
y = x4 + 5x3 + 2x2 – 8x
1. The leading coefficient is positive and the degree is odd
2. The end-behavior of the graph is rising on both sides.
3. x-intercepts  -4, -2, 0, 1
y-intercept  0
4.
i.
X
-4
-3
-2
-1
0
1
2
3
y
0
-12
0
6
0
0
48
210
5.
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ii.
1. The leading coefficient is positive and the degree is odd
2. The end behavior of the graph on the left side is falling while rising on the right
side
3. x-intercepts  -2, -1/2, -3
y-intercept  6
4.
X
-4
-3
-2
-1
0
1
2
3
y
-14
0
0
-2
6
36
100
210
5.
LESSON 2: ASSESSMENT
GUIDED PRACTICE
1. Roots: 1, -1, 3, 2
Turning Points: 3
2. Roots: -5, -3, 1,
Turning Points: 2
3. Roots: 3, 1, 1/2,
Turning Points: 3
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INDEPENDENT PRACTICE
Give at least 2 out of 4 different reasons why the graph below cannot be the graph of
f(x) = x4 + x2 + 1
1.
2.
3.
The given polynomial has degree 4 and positive leading coefficient , hence the graph
should rise on the left and on the right side.
y- intercept is 1but the graph shows that it did not pass through (0,1)
The equation has no solution which suggests that the function f(x) = x4 + x2 + 1 has
no zeros, yet the graph shows x – intercepts.
INDEPENDENT PRACTICE
1.
3.
2.
4.
5.
POST TEST
1. A
2. D
11. B
12. A
3. B
13. C
4. C
14. C
5. C
15. C
6. C
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7. A
8. D
9. D
10. C
References
BURGER, EDWARD, CHARD, DAVID, ET AL. Algebra 2, United States of
America, 2007
CALLANTA, MELVIN, CANONIGO, ALLAN, ET AL. Mathematics 10 Learner’s
Module, Rex Book Store Inc., Sampaloc, Manila, Philippines, 2015.
DILAO, SOLEDAD, ORINES, FERNANDO, ET AL. Advanced Algebra,
Trigonometry and Statistics(Revised Edition) , SD Publications, Inc.,
Quezon City, Philippines 2009.
ORONCE, ORLANDO AND MENDOZA, MARILYN. 2007. E-Math IV(First
Edition), Rex Book Store Inc., Sampaloc, Manila, Philippines, 2007
DESMOS graphing calculator : https://www.desmos.com/calculator
Microsoft Mathematics Software
https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_al
g_tut35_polyfun.htm
Lesson Tutorial Videos:
 https://www.youtube.com/watch?v=CHEtGgTexHI
 https://www.youtube.com/watch?v=3ANqMj5cfY8
For inquiries and feedback, please write or call:
Department of Education –Learning Resources Management and
Development Center (LRMDC)
DepEd Division of Bukidnon
Sumpong, Malaybalay City, Bukidnon
Telefax:
((08822)855-0048
E-mail Address:
bukidnon@deped.gov.ph
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