DAILY
LESSON
LOG
Annex to DepEd Order 42,
s.2016
School
Grade
TEN (10)
Level:
Learning Area:
MATHEMATICS
Quarter:
3rd Grading
FR. GRATIAN MURRAY, AFSC INTEGRATED SCHOOL
Teacher
Teaching Dates & Time
I. OBJECTIVES
1. Content Standards
2. Performance Standards
3. Learning Competencies
Objectives
BEVERLY N. ALON
March, 2023
The learner demonstrates understanding of key concepts of
polynomial function.
The learner is able to conduct systematically a mathematical
investigation involving polynomial functions in different fields.
The learner graphs polynomial function. (M10AL-IIa-b-1)
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.
a.
Graphs of Polynomial Functions
II. CONTENT
III. LEARNING RESOURCES
A. References
pp. 93-105
1. Teacher’s Guide
pp. 112-121
2. Learner’s Materials
3. Textbook
4. Additional Materials from Learning
Resources (LR) portal
B. Other Learning Resources
IV. PROCEDURES
A. Reviewing previous lesson or
presenting the new lesson
B. Establishing a purpose for the
lesson
C. Presenting examples/Instances
of the new lesson
Find your Match
Group the class into 5. Give each group a polynomial function. Have
them match the assigned function to them to the given graphs on the
board.
The group who got the correct answer earns 5 points.
Aside from the Intercepts, there are many other things to consider
when we draw the graph of a polynomial function. 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
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 xaxis 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.
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
If the polynomial
function
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
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.
2. Describe the behavior of the graph of
𝑦 = 𝑥4 − 5𝑥2 + 4
a. x- and y-intercepts
The polynomial in factored form is
𝑦 = (𝑥 − 1)(𝑥 + 1)(𝑥 − 2)(𝑥 + 2)
The roots(x-intercepts) are 1,−1, 2 and −2.
The y-intercept is 4
The graph will intersect the x-axis at (−2,0),(−1,0),(2,0), (1,0) and the
y-axis at (0,4).
b. multiplicity
There are no roots of even multiplicity.
c. behavior of the graph:
𝑛 = 4 and is even
Since 𝑛 is even and 𝑎𝑛 >0, then the graph comes down from the
extreme left and goes up to the extreme right.
d. turning points
There are 3 turning points.
D. Discussing new concepts and
practicing new skills # 1
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.
E. Discussing new concepts and
practicing new skills # 2
Is it possible for the degree of function to be less than the
number of turning points?
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:
F. Developing mastery (leads to
Formative Assessment 3)
Describe the graph of the following polynomial functions:
1. 𝑦 = 𝑥3 + 3𝑥2 − 𝑥 − 3
2. 𝑦 = −𝑥3 + 2𝑥2 +
11𝑥 - 12
G. Finding practical application of
concepts and skills in daily living
GROUP ACTIVITY
Describe the graph of the following polynomial
functions:
1. 𝑦 = 𝑥3 − 𝑥2 − 𝑥 + 1
2. 𝑦 = (2𝑥 + 3)(𝑥 − 1)
(𝑥 − 4)
H. Making generalizations and
abstractions about the lesson
Things to consider before we draw the graph of a polynomial
function.
a. x- and y- intercepts
b. multiplicity of roots
If 𝑟 is a zero of odd multiplicity, the graph of (𝑥) crosses
the x-axis at r.
If is a zero of even multiplicity, the graph of (𝑥) is tangent
to the xaxis at 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
I. Evaluating learning
J. Additional activities for
application or remediation
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 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
A. Follow Up
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
B. Study:
Graph of the function 𝑦 = 𝑥6 + 4𝑥5 + 4𝑥4 − 2𝑥3−5𝑥2 −
2𝑥
Prepared by:
BEVERLY N. ALON
Teacher I
Checked by:
RICARDO CAMINIAN
Head Teacher 1