MATHEMATICS
10
MATHEMATICS
SECOND QUARTER
WEEK 1 – 8
Learning Competency
Week 1
✓ Illustrates polynomial functions.
Illustrate Polynomial Functions
Polynomial function is a function which involves only nonnegative integer powers only. It may be
written in the form of f(x) or y. Consider the first set of examples below.
𝑓 𝑥 =−𝑥3+5𝑥2+6𝑥−1
𝑓 𝑥 =−𝑥3+5𝑥-2+6𝑥−1
Based from the definition of a polynomial function, powers must be non-negative or positive only. The
first example is a polynomial function while the second is not because it has negative powers.
For better understanding of the terms in polynomial functions, consider the example: 𝒚=−𝟐𝒙3+𝒙2−𝟗𝒙−𝟓.
Leading Term – basically it is the term that contains the highest power of x. In the given example, -2x3
is the leading term.
Leading Coefficient – it is the numerical coefficient of the leading term. In the given example, -2 is the
leading coefficient.
Constant term – it is a term which does not contain any variable. In the given example, -5 is the
constant term.
Degree of the Polynomial – Just simply get the highest powers among the terms of x. In the given
example, 3 is the degree of the polynomial function.
How to write polynomial functions in standard form?
𝑦 = 3𝑥 + 5𝑥3 − 2𝑥2 + 𝑥4 + 6
✓ Simply rearrange the terms in decreasing powers of x. The first step to follow is to identify the
leading term. From the given function, the leading term is the highest among the terms of x. That
is, x4.
✓ After that, we identify the next terms with powers less than 4. These terms are 5x3, -2x2, 3x, and
6.
✓ Take note, a constant term is always written at the end. Therefore, the standard form is 𝑦 = 𝑥4 +
5𝑥3 −2𝑥2 +3𝑥 +6.
Activity Card
A. Study the following polynomial functions. Then, fill in the table below. Write your
answers in your yellow pad paper.
Polynomial Function
1. f (x) = x3 +5x2 +8x −10
1
2. f (x) =−2x4 +3x3 −5x2 −7
3. f (x) =−3x2 −56x +18
4. y =2x5 −x4 −8x3 −5x2 +x
5. y =4x3 −2x2 +11x −9
Degree
3
Leading
Term
x3
Leading
Coefficient
1
Constant
Term
-10
B. Rewrite the following polynomials in standard form. Write your answers in your yellow
pad paper.
Example: 𝑓(𝑥) =−𝑥6 −𝑥3 +7𝑥8 +12𝑥2 +9 , Standard Form = 7𝑥8 −𝑥6 −𝑥3 +12𝑥2 +9
1. 𝒚 = 𝟐𝒙− 𝟓𝒙4 +𝟏𝟏𝒙2 −𝟑
2. y = 𝟕𝒙 +𝒙2 −𝟐𝒙3 +𝟑𝒙4
3. 𝒇 (𝒙) = 𝟔𝒙4 +𝒙7 −𝟏𝟏 +𝒙2
4. f (x) = 𝟑𝒙3 −𝟖 +𝟏𝟎𝒙2 −𝒙5
5. f (x) = 4x2 – 2x3 + x4 - 10
Learning Competencies
Week 2
✓ Understand, describe and interpret the graphs of polynomial functions.
✓ Solves problems involving polynomial functions.
Polynomial Functions
Polynomial functions may be written in factored form and as a product of irreducible factors. A
polynomial is said to have irreducible factors when it can no longer be factored using coefficients that
are real numbers.
Consider the example below:
What is the factored form of y = x3 + 2x2 – 2x – 3?
You can use synthetic division in finding the factors of the polynomial function. List down the possible
factors of the constant term “-3” which are 1, -1, 3, and -3. To determine the factors, you have to get a
remainder of 0 if you use synthetic division method.
From the 4 possible factors, only “-1” gives a remainder of 0. If you will divide (x3 + 2x2 – 2x – 3) by
(x + 1), the quotient is x2 + x – 3. You cannot factor x2 + x – 3 simply because it is irreducible /
unfactorable.
So, the factored form of y = x3 + 2x2 – 2x – 3 is (x + 1) (x2 + x – 3).
In graphing functions, x and y intercepts are important. These are points which intersect the coordinate
axes. How do we determine the intercepts of the graph? See example below.
Find the intercepts of the graph of y = x3 + 6x2 + 3x – 10. To determine the x-intercept/s, set y = 0. It would
be easy if we will use the factored form which is
y = (x + 5) (x + 2) (x – 1).
0 = (x + 5) (x + 2) (x – 1) Equate y to 0.
After that, equate each factor to 0.
x + 5 = 0
x = -5, x + 2 = 0
x = -2, x – 1 = 0
x=1
The x-intercepts are -5, -2, and 1. With this, the graph will pass through (-5, 0), (-2, 0), and (1, 0).
In finding the y-intercept, simply set x = 0.
y = (x + 5)(x + 2)(x – 1)
y = (0 + 5)(0 + 2)(0 – 1) Substitute x = 0.
y = (5)(2)(-1) Multiply.
y = -10
The y-intercept is -10. Thus, the graph will also pass through (0, -10).
Getting the intercepts is not enough to determine the sketch of the graph. We need to consider the use
of table of values to obtain other points.
Let us use the example above which is y = (x + 5) (x + 2) (x – 1). To get the values of y, substitute the
assigned values of x below. For easy reference, list down the answers in ordered pairs.
x
-4
Y
Below are the solutions in getting the y-values.
-1
0
If x = -4, then
If x = -1, then
If x = 0, then
y = (x+5) (x+2) (x–1)
y = (x+5) (x+2) (x–1)
y = (x+5) (x+2) (x–1)
y = (-4+5) (-4+2) (-4–1)
y = (-1+5) (-1+2) (-1–1)
y = (0+5) (0+2) (0–1)
y = (1) (-2) (-5)
y = (4) (1) (-2)
y = (5) (2) (-1)
y = 10
y = -8
y = -10
Other points of the graph of the polynomial function are (-4, 10), (-1, -8), and (0, -10).
Roller Coaster
Suspension Bridge
In graphing polynomial functions, you may consider the use of the Leading Coefficient Test. This
test will help you to determine the end behaviors (left-hand and right hand) of the graph of polynomial
functions. Make sure to write the polynomial function in standard form. Otherwise, LCT will be
incorrect and most importantly the graph.
Types of Graph of Polynomial Functions using the Leading Coefficient Test
Using y = x3 + 6x2 + 3x – 10, let us identify the leading coefficient, degree, end behaviors of the
graph.
The leading coefficient is 1 and the degree is 3 which is an odd. For the end behaviors of the graph, it
falls to the left and rises to the right.
Activity Card
A. Identify the coordinates of each point represented by the letters of the phrase “MATH IS
FUN”. The first example is done for you. Write your answers in your yellow pad paper.
M
A
T
H
I
S
F
U
N
(-3, 5)
B. Find the intercepts of the graphs of the following polynomial functions.
Standard Form
y = x3 + 9x2 +26x + 24
y = x3 + 6x2 – 19x - 84
y = -x4 + 3x3 + 18x2 – 40x
Factored Form
y = (x + 2) (x +3) (x +4)
y = (x + 7) (x + 3) (x – 4)
y = -x (x + 4) (x – 2) (x – 5)
x - intercepts
y - intercepts
-2, -3, -4
24
Learning Competency
✓ Derives inductively the relations among chords, arcs, central angles and
inscribes angles.
Week 3
Chords, Arcs, Central Angles and Inscribed Angles
A circle is the set of all points that are fixed from a point called the center.
In naming a circle we use its center. In the figure given, the name of the
circle is O because that is the center.
A radius is a line segment from the center to any point around the circle.
Two or more radius is called radii.
A diameter is a line segment that passes through the center and whose endpoints are on the circle.
Also called as the longest chord.
A chord is a line segment whose endpoints are on the circle. Unlike diameter, it does not pass
through the center of the circle.
An arc is a part of a circle. The symbol for arc is . One complete rotation of a circle is equivalent to
360°. Arcs measurement are expressed in degrees like angles. In naming arcs, two endpoints and
another point on the arc. There are three types of arc namely semicircle, minor arc, and major arc.
Consider the figure below.
An angle is a figure formed by two non-collinear rays. We name angle using the vertex.
In circles, we can form the following angles:
✓ A central angle is an angle formed by two rays whose vertex is the center of the circle.
✓ An inscribed angle is an angle formed whose vertex is on the circle and whose sides contain
the chords of the circle.
✓ An intercepted arc lies in the inner portion of an inscribed angle and whose endpoints is on the
angle. Also, the measure of a central angle is equal to the measure of the intercepted arc.
Inscribed Angle
Central Angle
The vertex of the angle is Point A.
So, we can also name it as ∠A.
In naming an angle, the letter representing the
vertex should be written at the middle.
Thus, we can only name it in two ways ∠DOG or
∠GOD.
Degree Measure of Arcs and Angles
Take note: The sum of all the central angles in a circle is 360°.
Activity Card
A. Find all the words you have learned in the lesson and encircle them. In this puzzle, there
are 10 words that you are going to hunt. Write your answers in your yellow pad paper.
B. Refer to the figure at the right to identify and name the terms related to the given circle
⊙ O.
1. A radius
2. A diameter
3. A chord
4. A semi – circle
5. A minor arc
6. A major arc
7. 2 central angles
8. 2 inscribed angles
Performance Task # 1
Parts of The Circle
Name and label at least 8 parts of the circle.
Scoring Rubrics
Learning Competency
✓ Proves theorems related to chords, arcs, central angles and inscribed
angles.
Week 4
Theorems related to Chords, Arcs, Central Angles and Inscribed Angles
Congruent Circle
Congruent Arcs
✓ In a circle or in congruent circles, two minor arcs are congruent if and only if their
corresponding central angles are congruent.
✓ In a circle or in congruent circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
✓ In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is
perpendicular to the chord.
✓ In a circle, if an angle is inscribed to it, then half the measure of the intercepted arc is the
measure of the inscribed angle.
✓ In a circle, the inscribed angles subtended by the same arc are equal.
✓ In a circle, if an angle inscribed to it intercepts a semicircle, it forms a right angle which
measures exactly 90°.
✓ An inscribed figure is drawn inside a polygon. While, a circumscribed figure is drawn outside a
polygon.
✓ In a circle, if a quadrilateral is inscribed to it, then its opposite angles are supplementary.
Supplementary angles are pairs of angles whose sum is 180°. There are two pairs of
supplementary angles in a quadrilateral.
Arcs Addition Postulate
✓ Postulate is a statement that is accepted
without prior proofs.
✓ Arc Addition Postulate states that the
arc formed by two adjacent arcs is the
sum of the measure of the two arcs.
Activity Card
A. Use the 3 different figures below to answer the questions. Write your answers in your
yellow pad paper.
B. Complete the following blank spaces in order to prove theorems on arcs, chords, central
angles and inscribed angles. Write your answers in your yellow pad paper.
Performance Task # 2
Inscribed Me!
Perform the activity below by following the procedures.
1. First, draw a circle. Mark and label the center of the circle as point E.
2. Second, draw a diameter of the circle. Label the endpoints as D and W.
3. Third, from the center of the circle, draw radius EL.
4. Fourth, draw angle LDW by connecting L and D with a line segment.
5. Lastly, name and describe the angle form.
Learning Competency
✓ Illustrates secants, tangents, segments, and sector of a circle.
Week 5
Sector and a Segments of a Circle
✓ A sector is like a slice of a pizza. It may be defined as a part of a circle enclosed by two radii
and an arc.
✓ A segment of a circle is the region bounded by an arc and the segment joining its endpoints.
Segments of a circle is different from sectors. If you cut a slice of pizza in a shape of triangle,
the portion cut off is like a crust of pizza.
Tangents and a Secant of a Circle
✓ If a line intersects a circle in exactly one point, then it is tangent line. The intersection point of
the circle and the tangent line is called point of tangency. A tangent line does not pass through
the interior of any given circle.
✓ When a line is tangent to two circles, it is considered as common tangents. There are two types
of common tangents namely common internal tangents and common external tangents.
✓ Common internal tangents are tangent lines that intersects segments joining the centers of
two circles.
✓ Common external tangents are tangent lines that do not intersect segments joining the centers
of two circles.
✓ In a circle, if a line intersects at exactly two points is called a secant line. Also, a secant line
contains chord of a circle. If a secant line intersects a circle in three points, then it intersects a
diameter.
Segments of Intersecting Chords and a Tangent
✓ Intersecting chords are two chords of the same circle that intersects a particular or sometimes
at the center. If two chords intersect at the center, then these chords are diameters of the given
circle.
✓ A tangent segment is a line segment that touches a circle at one point in particular.
✓ A secant segment is a line segment whose one endpoint is on the circle while the other endpoint
is outside the circle known as external secant segment.
Examples:
Activity Card
Refer to the colored figures to identify what is written in the colored box. Write your answers
in your yellow pad paper.
Performance Task # 3
Illustrate Me!
Perform the task below by following the procedures.
1. First, draw a circle with a center of P.
2. Second, put an inscribed square and name it as LMNO.
3. Third, draw two intersecting diameters. Connect Point L to Point N and from Point M to Point
O.
4. Fourth, identify the four congruent sectors.
5. Fifth, identify four segments.
Scoring Rubrics
Learning Competency
✓ Proves theorems on secants, tangents and segments.
Postulates and Theorems on Tangents
✓ Postulate is a statement that may be assumed as true without proofs.
✓ Theorem is a statement that has been proven to be true.
Week 6
Activity Card
There are 12 problems to be solved about angles formed by tangents and secants. Each box
corresponds to a word placed inside. Then, use the words to decode the message the unknown
measures are labeled question marks (?). Place the words opposite the answer. Write your
answers in your yellow pad paper.
Learning Competency
✓ Applies the distance formula to prove some geometric properties.
Week 7
Distance Formula
✓ The distance between any two point on a number line can be determined by the absolute value
of the difference between their coordinates.
Examples:
1. Find XY if 𝑋 (−4,5) and 𝑌 (1,2)
2. Show that the triangle whose vertices are A(-2,-2), B(6,-8) and C(4,6) is isosceles.
Midpoint Formula
✓ Measure the distance between the two end points, and divide the result by 2. This distance from
either end is the midpoint of that line. Alternatively, add the two x coordinates of the endpoints
and divide by 2. The results give you the coordinates of the midpoint.
Examples:
1. Locate the midpoint between (-3, -2) and (1, 0).
2. Locate the midpoint between (6, 3) and (6, 9).
Activity Card
A. Find the distance between each pair of points. Write your answers in your yellow pad
paper.
1. M (2, -3) and N (10, -3)
2. P (3, -7) and Q (3, 8)
3. C (-4, 3) and D (7, 6)
4. X (-3, 9) and Y (2, 5)
5. S (-4, -2) and T (1, 7)
B. Find the coordinates of the midpoint of the segment whose endpoints are given below.
Write your answers in your yellow pad paper.
1. A (6, 8) and B (12, 10)
2. C (5, 11) and D (9, 5)
3. K (-3, 2) and L (11, 6)
4. R (-2, 8) and S (10, -6)
5. P (-5, -1) and Q (8, 6)
Performance Task # 4
What figure am I?
Plot each set of points on the coordinate plane. Then connect the consecutive points by a line
segment to form the figure. Write your answers in your yellow pad paper.
1. A (6, 11), B (1, 2), C (11, 2)
2. L (-2, 8), I (5, 8), K (5, 1), E (-2, 1)
3. C (4, 12), A (9, 9), R (7, 4), E (1, 4), S (-1, -9)
4. B (1, 6), E (13, 7), A (7, -2), T (-5, -3)
5. S (-1, 5), O (0, -1), N (6, -6), G (-4, 0)
6. D (-4, 6), A (8, 6), T (8, -2), E (-4, -2)
Learning Competencies
✓ Illustrates the center – radius form of the equation of a circle.
✓ Determines the center and radius of a circle given its equation and vice versa.
Week 8
Equation of a Circle
✓ The standard equation of a circle with center at (h, k) and a radius of r units is (x – h)2 + (y – k)2
= r2. The values of h and k indicate that the circle is translated h units horizontally and k units
vertically from the origin. The general form a circle is x2 + y2 + Dx + Ey + F = 0.
✓ (h, k) is the coordinate of the center
✓ r is the radius of the circle
✓ D = -2h
E = -2
F = h2+k2- r2
Determine the standard form of the equation of a circle
given its center and radius.
Example: What is the standard form of the equation of a
circle with center at (2,7) and a radius of 6 units?
Determine the general form of the equation of a circle
given its center and radius.
Example: What is the general form of the equation of a circle with center at (2,7) and a radius of 6
units?
Determine the center and the radius of a circle given the standard form of its equation.
Example: Determine the center and the radius of a circle given the equation (x-2)2 + (y-7)2 =36.
Activity Card
Find the center and radius (in units) of a circle when the standard form of its equation.
A.(x+2)2 + (y-1)2 = 81
B.(x+6)2 + (y+1)2 = 25
2
2
C.(x+1) + (y-5) = 100
D.(x+7)2 + (y+10)2 = 16
E.(x-4)2 + (y+1)2 = 9
F.(x+2)2 + (y-2)2 = 36
2
2
G.(x+8) + (y+7) = 49
H.(x+3)2 + (y+6)2 = 81
I.(x-8)2 + (y-2)2 = 100
J.(x-6)2 + (y-9)2 = 16
Letter
A
B
C
D
E
F
G
H
I
J
Center
(-2, 1)
Radius
9