The document provides information about graphing polynomial functions, including: 1) How to determine the degree, leading coefficient, intercepts, and behavior of a polynomial function graph from its standard and factored forms. Activities are provided to match polynomial functions and determine intercepts. 2) How to use the leading coefficient test to determine if a polynomial graph rises or falls on the left and right sides based on whether the leading coefficient is positive or negative and if the degree is odd or even. Examples analyze the behavior of specific polynomial function graphs. 3) How to sign a table to summarize the intercepts, degree, leading coefficient, and behavior of polynomial function graphs. Students are asked to graph specific functions and

1. Republic of the Philippines
Department of Education
Region III
Division of Zambales
ILEEN C. MENES
Don Marcelo C. Marty High School

2. Least Mastered Skill
Graph polynomial functions.
Subtasks:
Determine leading coefficient and degree of
polynomial functions
Identify y-intercept and x-intercepts of the
graphs of polynomial functions in factored
form
Describe the behavior of the graph of
polynomial functions using Leading
Coefficient Test

3. A polynomial function is a function such as a quadratic, a
cubic, a quartic, and so on, involving only non-negative integer
powers of x. We can give a general definition of a polynomial, and
define its degree.
A polynomial of degree n is a function of the form
f(x) = anxn + an-1xn-1 + …+ a2x2 + a1x + a0
where the a’s are real numbers (sometimes called the coefficients
of the polynomial). Although this general formula might look quite
complicated, particular examples are much simpler. For example,
f(x) = 4x3 – 3x2 + 2
is a polynomial of degree 3, as 3 is the highest power of x in the
formula. This is called cubic function.
Let us investigate on how to determine and graph
polynomial functions. Let’s try if the following activities can help us
to graph polynomial functions easily.

4. Get Connected….
Match Column A (polynomial functions in standard form) to Column
B (polynomial function in factored form. Connect same polynomial
functions using a line.
A B
y = x4 – 26x2 + 25 ● ● y = - (x +3)(x –2)(x–4)(x+1)
y = 2x3 – 7x2 – 7x + 12 ● ● y = - (x-1) (x+1)2 (x – 2)2
y = -x5 + 3x4 + x3 - 7x2 + 4 ● ● y = (x+1) (x-1) (x+5) (x-5)
y = x4 – 7x2 + 6x ● ● y = x(x – 1)(x – 2)(x + 3)
y = -x4 +2x3 +13x2 – 14x -24 ● ● y = (x-1) (2x+3) (x-4)

5. Seize and Intercept Me….
Determine the intercepts of the graph given its polynomial
function in factored form.
1. y = (x-1) (2x+3) (x-4) x-intercepts: __, __, ___
y-intercept: __
2. y = - (x-1) (x+1)2 (x – 2)2 x-intercepts: __, __, ___
y-intercept: __
3. y = x(x – 1)(x – 2)(x + 3) x-intercepts: __, __, ___, __
y-intercept: __
4. y = -(x+3)(x–2)(x–4)(x+1) x-intercepts: __, __, ___, __
y-intercept: __
5. y = (x+1) (x-1) (x+5) (x-5) x-intercepts: __, __, ___, __
y-intercept: __
Key points:
You can determine x-
intercepts by finding
the roots of the
polynomial function,
y = -(x-1)(x+4)(x-5)
Roots:
x-1=0; x=1
X+4=0; x=-4
X-5=0; x=5
however y-intercept
can be determined
by multiplying all
second terms of the
linear factors
-(-1)(4)(-5) = -20

6. Did you miss me? Here I am again….
Consider the given polynomial functions and fill in the table below.
POLYNOMIAL FUNCTIONS
Degree
Leading
Coefficient(in standard form) (in factored form)
y = x4 – 26x2 + 25 y = (x+1) (x-1) (x+5) (x-5)
y = 2x3 – 7x2 – 7x + 12 y = (x-1) (2x+3) (x-4)
y = -x5 + 3x4 + x3 - 7x2 + 4 y = - (x-1) (x+1)2 (x – 2)2
y = x4 – 7x2 + 6x y = x(x – 1)(x – 2)(x + 3)
y = -x4 +2x3 +13x2 – 14x -24 y = - (x +3)(x –2)(x–4)(x+1)

7. Case 1. The graph on the right is defined by
y = 2x3 – 7x2 – 7x + 12 or
y = (x-1) (2x+3) (x-4)
Questions:
a. Is the leading coefficient a positive or negative number
b. Is the polynomial of even or odd degree?
c. Observe the end behaviors of the graph on both sides.
Is it rising or falling to the right or o the left?
Case 2. The graph on the right is defined by
y = -x5 + 3x4 + x3 - 7x2 + 4 or
y = - (x-1) (x+1)2 (x – 2)2
Questions:
a. Is the leading coefficient a positive or negative number
b. Is the polynomial of even or odd degree?
c. Observe the end behaviors of the graph on both sides.
Is it rising or falling to the right or o the left?
Follow Me……
Examine 4 cases illustrated below
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
-7
-7 - 6 -5 -4 -3 -2-1
7
6
5
4
3
2
1
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
-7
-7 - 6 -5 -4 -3 -2-1
7
6
5
4
3
2
1

8. Case 3. The graph on the right is defined by
y = x4 – 7x2 + 6x or
y = x(x – 1)(x – 2)(x + 3)
Questions:
a. Is the leading coefficient a positive or negative number
b. Is the polynomial of even or odd degree?
c. Observe the end behaviors of the graph on both sides.
Is it rising or falling to the right or o the left?
Case 4. The graph on the right is defined by
y = -x4 +2x3 +13x2 – 14x -24 or
y = - (x +3)(x –2)(x–4)(x+1)
Questions:
a. Is the leading coefficient a positive or negative number
b. Is the polynomial of even or odd degree?
c. Observe the end behaviors of the graph on both sides.
Is it rising or falling to the right or o the left?
(Continuation of Activity 4)
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
-7
-7 - 6 -5 -4 -3 -2-1
7
6
5
4
3
2
1
1 2 3 4 5 6 7-3
-6
-9
-12
-15
-6 -
-21-
-24
-7 - 6 -5 -4 -3 -2-1
3
2
1

9. Sign on and Sketch me….
For each of the following polynomial functions, sign on the table.
POLYNOMIAL FUNCTIONS
y-
intercept
x-
intercepts
Degree
(odd or
even)
Leading
Coefficient
(n>0 or
n<0)
Behavior of
graph
(rising/falling
)
Possibl
e
Sketch
(in standard form) (in factored form) Left-
hand
Right
-hand
y = x4 – 26x2 + 25 y = (x+1) (x-1) (x+5) (x-5)
y = 2x3 – 7x2 – 7x + 12 y = (x-1) (2x+3) (x-4)
y = -x5 + 3x4 + x3 - 7x2 + 4 y = - (x-1) (x+1)2 (x – 2)2
y = x4 – 7x2 + 6x y = x(x – 1)(x – 2)(x + 3)
y = -x4 +2x3 +13x2 – 14x -24 y = - (x +3)(x –2)(x–4)(x+1)

10. It’s your turn, show me….
For each of the following polynomial functions, give
a.) intercepts
b.) leading coefficients and degree
c.) behavior of graph of function using Leading Coefficient Test
d.) possible sketch of the graph
e.) graph of the polynomial function
1. y = x3 + 4x2 – 15x -18
or in factored form
y = (x+1) (x-3) (x+6)
2. y = - x4 + 13x2 -36
or in factored form
y = - (x+2) (x-2) (x+3) (x-3)

11. (Sample Graphing Paper)

12. Make Me Useful by Producing Something
Read the problem carefully and answer the question that
follow.
Because of the upcoming All Souls Day, you
are designing candle-making kits. Each kit contains 25
cubic inches of candle wax and a mold for making a
pyramid shaped candle with a square base. You want
the height of the candle to be 2 inches less than the
edge of the base
Questions/Tasks:
1. What should the dimensions of your candle mold be?
Show a mathematical procedure in determining the
dimensions.
2. Use a sheet of cardboard as sample material in
preparing a candle mold with such dimensions. The
bottom of the mold should be closed. The height of
one face of the pyramid should be indicated
3. Write your solution in one of the faces of your output
(mold)
How are
polynomial
functions used in
solving real-life
problems?

13. Intercepts and the Leading Coefficient Test
helped you a lot in graphing polynomial functions.
Also, we can use the concept of these to solve real-
life situations.
Summary:
If the degree is odd and the leading coefficient
is greater than 0, the graph will fall to the left and will
rise to the right. On the contrary when leading
coefficient is less than 0, it will rise to the left and fall
to the right. But if the degree is even, the graph will
both fall and rise on both sides depending on the
value of the leading coefficient.

14. Activity 1.
y = x
4
– 26x
2
+ 25 ● ● y = - (x +3)(x –2)(x–4)(x+1)
y = 2x3 – 7x2 – 7x + 12 ● ● y = - (x-1) (x+1)2 (x – 2)2
y = -x5 + 3x4 + x3 - 7x2 + 4 ● ● y = (x+1) (x-1) (x+5) (x-5)
y = x4 – 7x2 + 6x ● ● y = x(x – 1)(x – 2)(x + 3)
y = -x4 +2x3 +13x2 – 14x -24 ● ● y = (x-1) (2x+3) (x-4)
Activity 2.
1. [1, -3/2, 4] 3. [0, 1, 2, -3]5. [-1, 1, -5, 5]
[12] [0] [25]
2. [1, -1, 2] 4. [-3, 2, 4, -1]
[4] [-24]
Activity 3.
Degree Leading Coefficient
1. 4 1
2. 2 3
3. 5 -1
4. 4 1
5. 4 -1
Activity 4.
1. a. positive
b. odd
c. falling to the left and rising to
the right
2. a. negative
b. odd
c. rising to the left and falling to
the right
3. a. positive
b. even
c. rising to the left and to the
right
4. a. negative
b. odd
c. falling to the left and to
the right
Activity 5. (refer o Activity nos. 2-4)
1 2. 3. . 4. 5.

15. Assessment:
1. a) x-intercept: -1, 3, -6 2. a) x-intercept: -2, -3, 2, 3,
y-intercept: -18 y-intercept: --36
b) leading coefficient: 1 (>0) b) leading coefficient: -1 (<0)
degree: 3 (odd) degree: 4 (even)
c) falling(left), rising (right) c) falling(left), falling (right)
d) d)
e) e)
1 2 3 4 5 6 7-3
-6
-9
-12
-15
-18-
-21-
-24
-7 - 6 -5 -4 -3 -2-1
3
2
1
1 2 3 4 5 6 7-3
-6
-9
-12
-15
-18-
-21-
-24
-7 - 6 -5 -4 -3 -2-1
3
2
1
Enrichment:
Given : V = 1/3Bh then x = 5
V= 25 in3 25 = 1/3 [(x)2(x – 2)]
Let x be the edge of the square base 75 = x2(x-2)
then the height will be x-2 f(x) = x3 - 2x2 - 75 or y= (x-5) (x2 + 13x +15)

16. Cubic Function - a polynomial function whose degree is 1
Intercepts of a Graph – points on the graph hat have zero as either the x-coordinate or y-
coordinate
Leading Coefficient Test – a test that uses the leading term of the polynomial function to determine
the right hand and the left-hand behaviors of the graph
Linear function – a polynomial function whose degree is 1
Nonnegative Integer – zero or any positive integer
Polynomial Function – a function denoted by
f(x) = anxn + an-1xn-1 + …+ a2x2 + a1x + a0
Polynomial in Factored form – any polynomial represented by product of irreducible factors
Polynomial in Standard Form – any polynomial whose terms are arranged in decreasing powers of
x
Quadratic Function - a polynomial function whose degree is 2
Quartic Function - a polynomial function whose degree is 4

17. References:
Mathematics – Grade 10 Learner’s Module, First
Edition 2015 pp. 106-125.
Oronce, Orlando A. General Mathematics, First
Edition. Rex Printing Company, Inc., 2016.
Web Links:
www.mathcentre.ac.uk
www.youtube.com
www.mathsisfun.com

18. I am ready for another test (71-90)
I still need to recap the lesson (51-70)
I still need full remediation (50&below)
Score:
_____
_
90

19. (Sample Graphing Paper)