Topic: Quadratic Functions
Sub-Topic: Graph of Quadratic Functions
OBJECTIVES
a. Analyze the effect of different combinations of a, h and k on the graph of a quadratic
function
b. Predict the graph of a quadratic function given the values of a, h and k
c. Value accumulated knowledge as means of new understanding
1
Given
a.
b.
c.
d.
, answer the following questions.
Does the graph open upward or downward?
Is the vertex located to the left or to the right of the y-axis? By how many units?
Is the vertex located above or below the x-axis? By how many units?
Graph
.
Let’s Do More!
Given the quadratic functions, identify the values of
then draw their graphs.
a.
b.
c.
d.
travelled when it reaches its maximum vertical distance?
Ranidel de Ocampo of Talk „n
Text Tropang Texters made a
jumpshot. The height the ball
reaches (in inches) is given
by the equation
, where
is the
distance it had travelled in
inches.
a. What is the maximum
vertical height that the ball
can reach?
b. What horizontal distance
would the ball have
2
I Can Do This!
Quadratic Functions
Sub-Topic: Finding the Equation of a Quadratic Function
OBJECTIVES
a. Determine the equation of quadratic functions
b. Find the equation of a quadratic function given its vertex and one point on the graph
c. Find the equation of a quadratic function given its roots
d. Find the equation of a quadratic function given a set of ordered pairs
e. Value accumulated knowledge as means of new understanding
Things to Remember:
𝑥
The graph of a quadratic function is a parabola, which may cross the x-axis
once, twice or not at all. The x-coordinates of these points of intersection are called
-intercepts. The quadratic function can also be derived from a given set of points on the
graph.
If the vertex of the parabola is given, we can use the vertex form of the quadratic function.
If given the vertex and at least𝑟 one point,
𝑟 then use the vertex form of the quadratic
function. If given the roots and
and
another
point,
then
𝑓 𝑥
𝑎 𝑥 𝑟 𝑥 𝑟 the quadratic
𝑎
function is defined by
where is determined by substituting
the given point.
If given any three points or more points, the solve by making a system of linear equations by
substituting the values to the equation and
𝑦 𝑎𝑥
𝑏𝑥 𝑐
solving for the variables 𝑎 𝑏
𝑐
Let’s Do This!
3
Topic:
The set of points below correspond to a quadratic function.
x
-1
1
2
3
y
-5
3
1
-5
a. Substitute any three ordered pairs in the equation
b. What are the three equations that you came up with?
.
_______________, _______________, ______________
c. Using the three equations, solve the values of
d. What is the equation of the quadratic function?
y
4
Find the equation of the quadratic function
whose graph is shown.
3
2
1
x
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
Let’s Do More!
The vertex of a parabola is
and it passes through
the quadratic function which represents the parabola?
. What is the equation of
Challenge Yourself!
A volleyball launched from
followed a parabolic
path and touched the net located at
but
continued on its original path and
ended up in
. Find an equation which describes
the path taken by the volleyball.
Quadratic Functions
4
I Can Do This!
Sub-Topic: Solutions to Problems Involving Quadratic Functions
OBJECTIVES
a. Identify the solution of quadratic function
b. Solve problems involving quadratic function
c. Apply concepts learned in solving problems involving quadratic function
Things to Remember:
Quadratic function can be applied in different fields like physics, industry, business and in various
other mathematical problems. Familiarity with quadratic function, their zeros and their properties
is very important in solving real
-life problems.
As with any other algebraic problems, we always represent all unknowns with the
use of variables. We then construct a mathemat ical model that describes to the problem. We
must always note what each of the variables pertain to, to avoid
confusion.
Let’s Do This!
A. A tennis ball is thrown directly upward on top of a 119 ft. Maliksi-type building.The height
of the ball above the building after seconds is given by the function
.
a. What maximum height will the tennis ball reach?
b. How long will it take the tennis ball to reach the maximum height?
c. Find the time at which the tennis ball is on the ground.
B. A painting of Juan Luna 8 in. by 12 in. is placed in a frame which has a uniform width. If
the area of the frame is equal to the area of the picture, find the width of the frame.
I Can Do This!
5
Topic:
A. A ball is thrown across a playing field. Its path is given by the equation
, where is the distance the ball has travelled horizontally, and
its height above the ground level, both measured in feet.
a. What is the maximum height attained by the ball?
b. How far would it have travelled when it hits the ground?
is
6
B. A rectangular swimming pool at Villa Colmenar is 10m long and 12 m wide. A walk of
uniform width is to be built around the pool using 104m2 of tile. How wide should the walk
be?
Let’s Do More!
A. The effectiveness of a television commercial depends on how many times a viewer watches
it. After some experiments, an advertising agency found that if the
effectiveness is measured on a scale of 0 to 10, then
, where is the
number of times a viewer watches a given commercial. For a commercial to have maximum
effectiveness, how many times should a viewer watch it?
B. The number of guyabano fruits produced by each tree in a guyabano orchard depends on how
densely the trees are planted. If trees are planted on a hectare
land, then each tree produces
guyabano fruits. So the number of guyabano fruits
produced per hectare is
. How many trees should be planted per hectare
in order to obtain the maximum yield of guyabano fruits?
Challenge Yourself!
A kalamay Buna manufacturer finds that the revenue generated by selling units of wellpackaged kalamay is given by the function
, where the revenue
is
measured in pesos.
a. What is the maximum revenue?
b. How many units should be manufactured to obtain the above revenue?
Variation
Sub-topic: Direct Variation
OBJECTIVES
a. Illustrate situations that involve direct variation
b. Identify examples of direct variation
c. Appreciate the concept of direct variation in real-life situation
7
Topic:
Things to Remember:
There is a direct variation whenever a situation produces pairs of numbers in which
their ratio is constant.
The statements:
“y varies directly as x”
“y is directly proportional to x” and
“y is proportional to x”
May be translated mathematically as y = kx, where k is the constant of variation.
For two quantities, x and y, an increase in x causes an increase in
Similarly, a decrease in x causes a decrease in y.
y as well.
Let’s Do These!
Ronnie walks a distance of 1 km per 20 min. at a constant rate. The table shows the distance
he has walked at a particular length of time.
distance in km (d)
1
2
3
4
time in minutes (t)
20
40
60
80
Questions:
a. What happens to the distance as the length of time increases?
b. If Ronnie walks 1.5 km in going to school, how much time did he spend in
walking?
c. What mathematical statement can represent the relation?
d. What mathematical operation did you apply in this case?
8
e. Is there a constant number involved?
I Can Do These!
Put a check ( ∕ ) on the blank before the letter if the given situation is a direct variation and
put a cross( x ) if it is not.
_____ a. The distance an airplane flies to the time travelling
_____ b. The time a teacher spends in checking test papers to the number
of students
_____ c. The number of hours to finish a job to the number of persons
working
_____ d. The age of a used cellphone to its resale value
_____ e. The number of persons sharing a buko pie to the size of the
slices of the pie
Let’s Do More!
Answer the following questions.
a. Rody sold 3 baskets of banana at Php 35 per kg. If a basket contains 8 kg,
how much did Rody earn?
b. Candies are sold at Php1.50 each. How much will a bag of 420 candies cost?
c. A photocopy machine can finish 500 pages in 3 minutes. How many pages
can the machine copy in 1 hour?
Challenge Yourself!
Every week, Lemuel puts Php 15.00 in his piggy bank. In the following table, n is the number
of weeks and s is the savings in peso.
n
s
1
15
2
30
3
45
4
60
5
75
6
90
a. What happens to Lemuel‟s savings as the number of weeks doubled?
b.
c.
d.
e.
tripled?
In how many weeks would he have saved Php 360?
How much will be his savings after 15 weeks?
What mathematical statement can represent the relation?
Give the constant number involved in the situation.
Variation
Sub-topic: Direct Variation
OBJECTIVES:
a. Translate into variation statement a relationship involving direct variation between two
quantities given by a table of values, a mathematical equation, and a graph, and vice
versa
9
Topic:
b. Find the unknown in a direct variation equation
c. Appreciate the concept of direct variation in real-life situation
Things to Remember:
There is a direct variation whenever a situation produces pairs of numbers in which
their ratio is constant.
The statements:
“y varies directly as x”
“y is directly proportional to x” and
“y is proportional to x”
May be translated mathematically as y = kx, where k is the constant of variation.
For two quantities, x and y, an increase in x causes an increase in
Similarly, a decrease in x causes a decrease in y.
y as well.
Let’s Do These!
In each of the following, find the constant of variation (k) and an equation that
defines the relation. a.
a
2
3
4
5
6
b
8
12
16
20
24
b.
10
80
70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
no. of students (n)
c. If p varies directly as q, and p = 25 when q = 20
d. If y varies directly as and y = 24 when x = 3
e. If a varies directly as the square of b and a =3 when b = 4
I Can Do These!
A. Express the following as an equation where k is the constant of variation.
a. The circumference (C) of a circle varies directly as the length of its
b.
c.
d.
e.
diameter (d).
The cost (C) of a fish varies directly as its weight (w) in kilograms.
The water bill (B) varies directly as the number of cubic meter
consumption (c).
The area (A) of a square varies directly as the square of a side (s).
The volume of (V) of a cylinder varies directly as its height (h).
B. Find k, if y varies directly as x and
a. x = 15 when y = 120
b. y = 45 when x = 5
c. x = 8 when y = 112
d. x = 9 when y = 21
e. y =14 when x = 42
Let’s Do More!
Answer the following questions.
a. If n varies directly as m, and n = 40 when m = 16, find n when m = 30.
b. If y varies directly as x, and y = 35 when x = 14, find y when x = 16.
11
c. If y varies directly as x, and y = 45 when x = 2 ½, find x when y = 180.
d. If p varies directly as q, and p = 21 when q = 24, find p when q =56.
e. If x varies directly as a + b, and x = 12 when a = 1 and b = 2, find x
when a = 3 and b = 4.
Challenge Yourself!
Find the variation constant and the equation of variation in each of the following if
y varies directly as x.
a. y = 15 and x = 3
b. y =48 and x = 6
c. y =10 and x = 24
d. y = 1 and x =
e. y =7.2 and x = 2
12
Topic: Variation Subtopic:
OBJECTIVES
Direct Variation
:
a. Solve problems involving direct variation
b. Apply the concepts learned about direct variation in solving verbal problems
c. Appreciate the concept of direct variation in real-life situation
Things to Remember:
Direct variation problems can be solved using the equation
y= kx
where k is the constant of variation
When dealing with word problems, we can consider using variables other than x and y.
Let’s Do These!
Activity:
Teacher Arlene charges Php 300 per hour of tutorial service to a Grade 7 student.
a. Complete the table if c is the amount charge for the tutorial service and n is the number of hours.
n
1
2
3
4
5
6
13
b.
c.
d.
e.
c
300
After completing the table, observe the values of c and n. What happen to the amount charge
(c) when the number of hours (n) doubled or tripled?
What mathematical statement relates the two quantities n and c?
What is the constant k of variation? variation equation?
If Teacher Arlene spends 3 hours tutoring per day, how much
would she receive after 20
days?
I Can Do These!
Solve:
a. Express an equation where k is the constant of variation: the voltage
(V) in an electric circuit varies directly as the current (I). If I = 5 when
V=
110, find V when I = 12.
b. The circumference of a circle varies directly as its diameter. If the variation
constant (k) is what is the circumference of a circle whose diameter is 12
cm? 18 cm? 25 cm?
Let’s Do More!
Solve:
a. If c varies directly as the square root of d, and c = 16 when d = 64, find c
when d = 324.
b. A student earns Php 280.00 for a 5 hour shift on his summer job. The amount
of money he earns varies directly to the number of hours worked. Find the
constant k, and the amount of money he will earn for an 8-hour shift.
c. The number N of bottles produced by a machine varies directly to the amount
of time t the machine is working. If the machine produces 15,000 bottles in 6
hours, how many bottles can it produce in 18 hours?
Challenge Yourself!
Solve:
a. The number of kilograms of water K in a human body is directly proportional
to its total weight W. A man who weighs 87 kg contains 58 kg of water. If
Marcelo weighs 72 kg, how many kilograms of water does he contain?
b. The service fee f of a professional physical therapist varies directly as the
number of hours h of service rendered. If a physical therapist charged Php
14
Topic: Variation Subtopic:
OBJECTIVES
1350 for 3 hours of service, how much would he be paid for 5 ½ hours of
service?
Inverse Variation
:
a. Illustrate situations that involve inverse variation
b. Translate into variation statement a relationship involving inverse variation
c. Appreciate the concept of inverse variation in real-life situation
Th to Remember:
Things
Inverse variation occurs whenever situations produce pairs of numbers whose
product is constant.
For two quantities x and y, an increase in x causes a decrease in y or vice
versa. We can say that y varies inversely as x or y = 𝑘 .
𝑥
The statement, “y varies inversely as x,” translate to
y = . where k is the constant of variation
𝑘
𝑥
Let’s Do These!
The statement “y varies inverse as x” is y = . Express each of the following as equation.
a. The number of slices (s) that can be made from a standard Pinoy loaf of bread is
b.
c.
d.
e.
inversely proportional to the thickness (t) of a slice.
At a constant voltage, the electric current (I) varies inversely as the resistance (R).
The volume (V) of a gas at constant temperature varies inversely as the pressure (P).
The altitude (h) of a triangle with a constant area varies inversely as the base (b).
The time (t) required to travel a given fixed distance is inversely proportional to the speed
(r).
I Can Do This!
a. y =
b. k = xy
c. k =
15
d.
y
2
4
6
8
x
5
10
15
20
y
x
15
2
6
5
5
6
e.
10
3
Let’s Do More!
Find the constant (k) of variation and write the equation representing the relationship
them.
a. y varies directly with x. If y = -4 when x = 2
b. y varies inversely with x. If y = 40 when x = 16
c. y varies inversely with x. If y = 7 when x = -4
d. y varies inversely with x. If y = 15 when x = -18
e. y varies inversely with x. If y = 75 when x =25
between
Challenge Yourself!
Find the constant (k) of variation and write the equation representing the relationship
between the quantities in each of the following.
a.
y
3
6
12
24
x
8
4
2
1
y
0.5
1
1.5
2
x
6
3
2
1.5
b.
c.
16
Topic: Variation Subtopic:
OBJECTIVES
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
width(w)
d. y varies inversely as x, and y = 30 when x = 9
e. c varies inversely as d, and c =48 when d = 10
Inverse Variation
:
a. Translate into variation statement a relationship involving direct variation between two
quantities given by a table of values, a mathematical equation, and a graph, and vice
versa
b. Solve problems involving inverse variation
c. Appreciate the concept of inverse variation in real-life situation
17
Things to Remember:
Steps in solving inverse variation problem:
The statement, “ y varies inversely as x,” translate to
constant of variation.
Step 1: Set up the formula:
y=𝑘
Step 2: Find the missing constant k,
using the first set of data given.
k = xy
y = 𝑘 . where k is the
𝑥
𝑥
Step 3: Using the formula and the constant k,
find the missing in the problem.
Let’s Do These!
Write an inverse variation equation that relates x and y. Assume that y varies
inversely as x. Then solve.
a. If y = 10 when x = 7, find y when x = 5
b. If y = 21 when x = 10, find y when x = 4
c. If y = 5 when x = 5, find x when y = 2
d. Find the value of y when x = 5 if y = 8 and x = 10
e. Find the value of y when x = if y = 27 and x =
I Can Do These!
Solve for the indicated variable in each of the following.
a. If y varies inversely with x. If y = 40 when x = 16, find x when y = -5.
b. If y varies inversely with x. If y = 7 when x = -4, find y when x = 5.
c. If a varies inversely as b and a = 2 when b = 3, find a when b = 6.
18
Topic: Variation Subtopic:
OBJECTIVES
d. If w varies inversely as v and w = 12 when v = 8, find w when v = 6.
e. If y varies inversely as the square of x and y = 3 when x = 4, find y when x
= 16.
Let’s Do More!
Solve the following problem:
a. The number of hours (t) required to finish a certain job varies inversely as the
number of persons (n) on the job. If 8 persons require 9 hours to finish the
job, how long should it take for 24 persons?
b. The bases (b) of triangles having equal areas are inversely proportional to
their altitudes (h). The base of a certain triangle is 12 cm and its altitude is 15
cm. Find the base of a triangle whose altitude is 20 cm.
c. The force of attraction between two opposite electrical charges varies
inversely as the square of the distance between them. If the force (F) = 18
when the distance (d) = 10, find F when d = 15.
Challenge Yourself!
Find the missing variable:
a. y varies inversely with x. If y = -4 when x = 2, find y when x = -6.
b. y varies inversely with x. If y = 20 when x = 8, find x when y = -5.
c. y varies inversely with x. If y = 7 when x = -4, find y when x = 5.
d. y varies inversely with x. If y = 15 when x = -18, find y when x = 27.
e. y varies inversely with x. If y = 75 when x =25, find x when y = 25.
Joint Variation
:
a. Translate into variation statement a relationship involving joint variation between
two quantities given by a mathematical equation and vice versa. b. Solve problems
involving joint variation.
c. Appreciate the concept of joint variation in real-life situation.
19
Things to Remember:
Joint Variation is the same as direct variation with two or more variables.
The statement “ a varies jointly as b and c ” means a = kbc, or
constant of variation.
k=
𝑎
𝑏𝑐
, where k is the
Let’s Do These!
Solve for the value of the constant k of variation, then find the missing value.
a.
If y varies jointly as the product of x and z, and y = 105 when x = 5 and z = 7,
find y when x = 9 and z = 10.
b.
If y varies jointly as the product of x and z, and y = 1000 when x = 10 and z = 20,
find y when x = 8 and z = 10.
c.
A varies jointly with l and w, when A = 24, l = 3 and w = 2. Find A when l = 12 and
w = 7.
I Can Do These!
Solve.
a. y varies jointly as x and z, find y if x =3, k = 6 and z = 9
b. c varies jointly as a and b. If c = 45 when a = 15 and b = 3, find c when a = 21
= 8.
c. m varies jointly as n and p. If p = 4 when m = 72 and n = 2, find p when m = 12
and n = 8.
d. q varies jointly as g and the square of b. If q = 105 when g = 14 and b = 5, find q
and b = 14
e. y varies jointly as x and z. If y = 20 when x = 4 and y = 3, find y when x = 2 and
3.
and b
when g = 10
y=
Let’s Do More!
Solve the following joint variation problem.
20
Topic: Variation Subtopic:
OBJECTIVES
a. If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4, find f
when g = 3 and h = 6.
b. If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c =
36,
find a when b = 8 and c = 225.
c. The volume (V) of a pyramid varies jointly as its height (h) and the area (A) of
its base. A
pyramid with a height of 12 feet and a base with area of 20 square
feet has a volume of 80
cubic feet. Find the volume of a pyramid with a height
of 17 feet and a base with an area of
27 square feet.
Challenge Yourself!
Solve the following.
a. r varies jointly as p and q and r = 30 when p = 5 and q = 3
i.
Find r when p = 8 and q = 7
ii.
Find p when r = 36 and q = 2 iii. Find q when r =
40 and p = 4
b. F varies jointly with D and E. When F = 98, D = 2 and E = 7. Find F when D = 15 and E = 8
c. The strength (S) of the rectangular beam varies jointly as its width (w) and the
square of its depth (d). If S = 1200 pounds per square inch and w = 3 inches
and d
= 10 inches, what is the strength of a beam four inches wide and 6
inches deep?
21
Topic: Zero Exponents, Negative Integral Exponents, Rational Exponents, and
Radicals
Sub-Topic: Zero Exponents, Negative Integral Exponents
Objectives:
1.Write the given expressions into its simplest form.
2.Learn and understand the key concepts of negative integral, zero ,
3.Apply the laws involving positive integral exponents to zero and negative exponents
22
Things to Remember:
Definition:
am x an = 𝑎𝑚
𝑛
Definition: Repeated multiplication
(am)n=amn
Definition: Fraction Raised to an Integral Power
Raising a Number to a Zero Exponent1
Definition:
a0=1
(a≠0)
Note 1: a0 = 1 is a convention, that is, we agree that raising any number to the power
0 is 1. We cannot multiply a number by itself zero times.
Note 2: In the case of zero raised to the power 0 (written 00), mathematicians have
been debating this for hundreds of years. It is most commonly regarded as having
value 1, but is not so in all places where it occurs. That's why we write a≠0.
Raising a Number to Negative Exponents
Definition
a−n =
𝑎𝑛
(once again, a≠0)
In this exponent r ule, a cannot equal 0 because you cannot have
bottom of a fraction.
0 on the
Let’s Do This!
Multiplying Expressions with the Same Base
23
Let's start with an example. Once you get the hang of this, it makes writing math a whole lot easier.
Say we need to multiply 2 large numbers,
to write:
and
. Now, if we write it out in full, we would need
=10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = (8 lots of 10 multiplied together)
=10 x
10 x 10 x 10 x 10 (5 lots of 10 multiplied together)
So:
x
= (10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 ) ×(10×10×10×10×10)
Now, if you count them all up, you will have 13 lots of 10 multiplied together.
So we can conclude that
x
= (10)8 + 5 = 1013
This is very tedious and there must be an easier way. We could add the exponents when multiplying
numbers with the same base. Let's see a general definition.
Definition: am x an =
Let's see how this works with an example involving a variable, b:
Example 2 b5×b3=(b×b×b×b×b)×(b×b×b)
Our final answer is equivalent to
Dividing Expressions with the Same Base
When we divide expressions with the same base, we need to subtract the exponent of the number
we are dividing by from the exponent of the first number. In general, we can write is as follows.
Definition: Dividing algebraic expressions
(of course, a≠0)
It may be easiest to see how this one works with an example.
Example 3
We cancel 2 of the b's from the numerator (the top) and the two b's from the denominator (the
bottom) of the fraction. The result is equivalent to b7−2.
24
We could also write this problem as
Repeated Multiplication of a Number Raised to a Power
Next we consider the case where we have a base raised to some exponent, then we raise that to
some other exponent.
For example, we may start with p3 and need to raise it to the power 2. How do we do that? We'll see
the answer in a minute. First, let's look at a general definition.
(
= (am) x (am) x (am) x….x(am)n
n
[We multiply n times]
So we write:
Definition: Repeated multiplication
(am)n=amn
Example 4
(p3)2 = p3 x p3=(p×p×p)×(p×p×p)=p6
A Product Raised to an Integral Power
In this section we have 2 numbers multiplied together, and we raise the result to some power. In this
case, it has the same value as raising the first number to the power and multiplying by the second
number raised to the power.
Definition: Product Raised to an Integral Power
(ab)n = anbn
Example 5
(5q)3 = 53q3 = 125q3
We have raised the 5 to the power 3 (giving us 125) and we can't do anything else with q3.
A Fraction Raised to an Integral Power
If we have a fraction raised to an integral power, we need to raise the top number to the power and
divide by the bottom number raised to the power.
Definition: Fraction Raised to an Integral Power
25
Example 6
In this example, I have written out in full the meaning of
raised to the power 4.
Example 7
Expand:
Answer: Raising the top and bottom numbers to the power of 5 gives:
Raising a Number to a Zero Exponent
Definition:
a0=1 (a≠0)
Example 8
70 = 1
Example 9
x0 = 1
Example 10
(5a)0 = 1
Note 1: a0 = 1 is a convention, that is, we agree that raising any number to the power 0 is 1. We
cannot multiply a number by itself zero times.
Note 2: In the case of zero raised to the power 0 (written 00), mathematicians have been debating this
for hundreds of years. It is most commonly regarded as having value 1, but is not so in all places
where it occurs. That's why we write a≠0.
Raising a Number to Negative Exponents
Definition
26
a−n =
(once again, a≠0)
In this exponent rule, a cannot equal 0 because you cannot have 0 on the bottom of a fraction.
Example 11
3-2 =
=
Example 13
xExplanation: 0 and Negative Exponents
Observe the following decreasing pattern:
34 = 81
33 = 27
32 = 9
31 = 3
What do you notice about each step. Now continuing beyond ,what do you notice?
Using a as base and m and n as exponent
I Can Do This!
Answer the following questions correctly and write your solution on your answer sheet. a. A
student can spend P10-2 in a day, how much will he spend in 104 days?
27
b. The weight of a newborn baby chicken weighs 3-2 grams. If an adult chicken can weigh up to 34 times
more than a newborn chicken, how much does an adult chicken weigh? Leave your answer in the form
a^b
c. The annual corn yield is 5a2 kg per hectare. If there are 2b8 hectares of corn field in Maragondon and
7b7 hectares of corn field in Magallanes, what is the total annual corn yield in these two cities?
d. A swimming center builds a new pool that has the dimensions 3a6 by a2 by b4. What is the volume of
the pool?
e. A Jollibee food company which sells beverage likes to use exponents to show the sales of the beverage
in a2 days. If the daily sales of the beverage is 5a4, what is the total sales in a2days?
Let’s Do More!
Try these practice problems, write the name of right icon appropriate to you answer
28
Challenge Your Self!
Apply laws of exponents in simplying the expressions below, write your solution on the blank
provided
a)
29
b)
c)
d)
e)
Topic: Zero, Negative and Rational Exponents
Sub-topic: Rational Exponent
OBJECTIVES
a. Define rational exponents
b. Illustrate expressions with rational exponents
c. Value accumulated knowledge as means of new understanding
THINGS TO REMEMBER:
A rational exponent is an exponent that can be exp ressed as 𝑚 where m and n
𝑛
are integers and n≠ . Radical expressions can be written by using rational
exponents. For any natural number n and integer m, the exponent indicates
𝑛
the nth root 𝑎𝑛
power
𝑎
𝑚
𝑛
𝑛
𝑛
𝑎𝑚
𝑎. The exponent
𝑛
𝑎
𝑚
𝑛
indicates the nth root raised to the mth
𝑚
Work in Pairs.
Fill up the table of the correct answer. One row is filled up as an example.
30
Column A
Column C
Column B
Value(s) of
that
satisfy the equation in
Column B
5 and -5
1.
2.
3.
4.
5.
6.
Find all real roots.
1. fourth roots of 81
2. cube roots of -125
3. sixth roots of -729
4. square roots of 100
5. fourth roots of 256
Simplify the following expressions, then decode the following
What’s This Farm???
This is located at Buho, Silang, Cavite. This farm is known for being the first
organic farm in the country. They manufacture local medicinal herbal teas made from
plants that are known to have healing properties. They also manufacture sauces,
dressings, dips and ready-to-eat food items. To find out what‟s this farm, write the letter
of the correct choice in each blank at the bottom that contains the exercise number.
1.
2.
D. 15
T. 2, -2
U. 5, -5
S. 4, -4
3.
R. -10
H.
8
F.
-9
4.
A. 1, -1
5.
6.
7.
E. 13, -13
O. 20, -20
N. -9
I.
___ ___
G. 3
___ ___ ___ ___ ___
6
M. undefined
31
1
2
3
4
1. In the expression
2. When we write
5
, we call
6
7
as a _________ exponent.
as a radical , n is the ________ of the radical
Select the expression from A, B and C that correctly completes the statement.
A
B
C
is equivalent to
4.
is equivalent to
5.
is equivalent to
Topic: Zero, Negative and Rational Exponents
Sub-topic: Rational Exponents
OBJECTIVES
a. Apply the laws involving integral exponents to rational exponents
b. Simplify expressions with rational exponents
c. Value accumulated knowledge as means of new understanding.
32
THINGS TO REMEMBER:
The laws of exponents for integral exponents may be used in simplifying
expressions with rational exponents.
Let m and n be rational numbers and x and y be real numbers, variables
or algebraic expressions.
1. 𝑥 𝑚 · 𝑥 𝑛 =𝑥 𝑚 𝑛
𝑚
2. 𝑥 𝑛 = 𝑥 𝑚 𝑛
𝑥
3. 𝑥𝑦 𝑚 = 𝑥 𝑚 𝑦 𝑚
4. 𝑥 𝑚 𝑛 = 𝑥 𝑚𝑛
𝑚
𝑚
5. 𝑥
= 𝑥𝑚
𝑦
𝑦
6. 𝑥 = 1
7. 𝑥 𝑚 = 𝑚
𝑥
8.
𝑥
𝑦
𝑚
𝑚
= (𝑦 𝑚 )
𝑥
Fill in the missing parts of the solution in simplifying expressions with rational exponents.
a.
b.
= ___
= ___
c.
d.
e.(
Simplify and express answers with positive exponents, then decode the following.
Is This Garden of EVE?
33
This garden is located at Barangay Buck State, Alfonso, Cavite which offers romantic
nature garden dining and lodging. The garden was supposed to be the owner‟s private
paradise and just opened the doors to the public in February 1998 in time for Valentine‟s
Day. What garden is this? To find out, write the letter of the correct choice in each blank
at the bottom that contains the exercise number.
1.
N.
2.
B.
A.
R.
3.
Y. 3
W. a
4.
C. 10
O.
5.
S.
I.
___ ___ ___ ___ ___
1
2
3
4
5
Using your knowledge of rational expressions, simplify the following
Given
Final Answer
1.
2.
3.
4.
5. (
Simplify the given expression
34
Topic: Zero, Negative and Rational Exponents
Sub-topic: Rational Exponents
OBJECTIVES
a. Relate expressions with a rational exponents to radical expressions.
b. Write expressions with rational exponents as radicals.
35
c.
Value accumulated knowledge as means of new understanding.
THINGS TO REMEMBER:
If 𝑚 is a rational number and a is a positive real number, then a𝑚 =
𝑛
𝑛
𝑚
m
𝑛
𝑛
𝑛
𝑎𝑚 = ( 𝑎𝑚 provided that 𝑎𝑚 is a real number. The form ( 𝑎) =a 𝑛 is
called the principal nthroot of 𝑎𝑚 .Through this we can write expressions
with rational exponents as
radicals.
𝑛
Carefully analyze the example below then fill in the rest of the exercises with the
correct answer.
1.
1.
3.
4.
36
Write each of the following expressions in radical form.
1.
2.
3.[(m+
4.[(x
5.
Rewrite in radical form, then decode the following
Fish Be With You!
It is a freshwater fish that is found in Taal Lake. Also called giant trevally, this fish
can weigh up to 3 kilos. It lookslike talakitok but tastes a lot better and best served when
grilled. What‟s this fish? To find out, write the letter of the correct choice in each blank at
the bottom that contains the exercise number.
1.
A.
B.
C.
√(
U.
N.
L.
P.
W.
I.
S.
6.
O.
7.
M.
8.
E.
T.
___ ___ ___ ___ ___ ___ ___ ___
√
G.
D.
37
1
2
3
4
5
6
7
8
Write each of the following expressions in radical form.
38
Topic: Zero, Negative and Rational Exponents
Sub-topic: Rational Exponent
OBJECTIVES
a. Relate radical expressions to expressions with rational exponents
b. Write radical expressions to expressions with rational exponents
c. Value accumulated knowledge as means of new understanding
THINGS TO REMEMBER:
If 𝑚 is a rational number and a is a positive real number, then a𝑚 =
𝑛
𝑛
𝑚
m
𝑛
𝑛
𝑛
𝑎𝑚 = ( 𝑎𝑚 provided that 𝑎𝑚 is a real number. The form ( 𝑎) =a 𝑛 is
called the principal nthroot of 𝑎𝑚 .
𝑛
Carefully analyze the examples below then fill in the rest of the exercises with
the correct answer.
1.
Write each of the following in exponential form.
1.
39
Write each as expressions with positive rational exponents then decode the following
Show Me Your Sweet Smile 
Amira‟s, Rowena‟s and Loumars are the most famous stop-over in Tagaytay for
pasalubong. What do you think is common to these Tagaytay‟s tourist flavour attraction?
To find out, write the letter of the correct choice in each blank at the bottom that contains
the exercise number.
1.
S.
2.
O.
P
3.
T.
A.
4.
I.
R.
E.
___ ___ ___ ___ ___
1
2
3
1
4
Write each as expressions with positive rational exponents.
1.
2.
3.
4.
5.
Topic: Zero, Negative and Rational Exponents
Sub-topic: Radicals
OBJECTIVES
a. Derive the laws of radicals
b. Simplify expressions applying the laws of radicals
c. Value accumulated knowledge as means of new understanding
40
THINGS TO REMEMBER:
The following are the laws on radicals
a. 𝑛√ 𝑥𝑦 𝑛 𝑥 𝑛√ 𝑦
b.
𝑛
𝑥
√𝑦
𝑚
𝑛
𝑥
𝑛
𝑦
𝑛
c.
√ 𝑥 𝑚𝑛 𝑥
d. x is a real number and n is odd, then 𝑛 𝑥 𝑛 𝑥
e. If x is a real number and n is even, then 𝑛 𝑥 𝑛
𝑥
.
Prove:
Proof:
Prove:
Proof: ____________
____________
____________
____________
Power Rule for Products
_____________
_____________
Simplify the following by applying the laws of radicals.
41
1.
2.
3.
4.
5.
Simplify each radical.
1.
2.
= _______
3.
Topic:Radicals Sub-topic:Simplifying Radicals
OBJECTIVES
a. State the laws on radicals
b. Simplify radical expressions by removing perfect nth powers
c. Appreciate the use of radicals in real life situation
42
THINGS TO REMEMBER:
The following are the different laws on radicals. Assume that when n is even,
a > 0.
n
𝑛
1.
𝑎) = a
𝑛
2.
𝑛
𝑛
𝑎𝑏 = 𝑎 ∙ 𝑏
𝑎
√𝑏 =
𝑛
3.
𝑛
𝑛
𝑎
𝑏
Product Rule for Radical
Quotient Rule for Radical
To simplify radicals by removing the perfect nth powers, break down the
radicand into perfect and non perfect nth powers and apply the property
𝑛
𝑛
𝑛
𝑎𝑏 = 𝑎 ∙ 𝑏.
Let’s Do This!
Simplify each of the following radical expressions.
1.
= ____
2.
Break down the radicand into perfect and nonperfect nth powers
Extract the perfect nth power
Apply law of radicals
3.
Extract the perfect nth power
=
=
Break down the radicand into perfect and
nonperfect nth powers
Extract the perfect nth power
43
Break down the radicand into perfect and
nonperfect nth powers
Extract the perfect nth power
I Can Do This!
Simplify the following radical equations. Then, fill up the next table with the letter that
corresponds to the correct answer.
What makes Caviteños sounded like Español?
B
A
O
N
5.
C
H
6.
Let’s Do More!
Solve the following.
A. Find the length of a side of a square whose area is:
1. 300
.
2. 250
.
3. 160 .
B. Find the edge of a cube whose volume is
4. 54 cu. cm.
5. 56 cu cm.
C. Solve.
6. Cavite, due to its close proximity in Manila,
has a remarkable increase in building housing
subdivisions. If a rectangular floor area of a
certain house in a
subdivision is 2000
square meters and the
width measures m, what
Challenge Yourself!
is its length?
Photo Credit: Google Image
44
Simplify.
1.
2.
3.
4.
5.
Topic: Radicals Sub-topic: Simplifying radicals
OBJECTIVES
a. Define rationalization
b. Simplify radicals by reducing the index to the lowest possible order and rationalizing the
denominator of the radicand
c. Determine whether the radical expression is in its simplified radical form d. Value
simplicity
45
THINGS TO REMEMBER:
Simplify Radicals by:
1. Reducing the index to the lowest possible order
Express the radical into an expression with a rational exponent then
𝑚
𝑛
simplify the exponent or apply the property √ 𝑛 𝑎 = 𝑚𝑛 𝑎 = √ 𝑚 𝑎 .
2. Rationalizing the denominator of the radicand
If there are radicals in the denominator, simplify the fraction into an
expression free of radicals in the denominator. This process is called
rationalizing the denominator.
A radical expression of index n is in simplified radical form if it has
1. no perfect nth powers as factors of the radicand,
2. no fractions inside the radical, and
3. no radicals in the denominator.
Let’s Do This!
Simplify. 1.
Apply the property
Find the root.
2.
Express into an expression with a rational exponent.
Reduce the exponent into lowest term.
Transform into radical form.
46
Remove the perfect nth power.
Reduce the index of the radical.
Rationalize the denominator
Remove the perfect square
Reduce to lowest term.
Quotient rule for radicals
Product rule for radicals
Simplify
Rationalize the denominator
I Can Do This!
Let‟s Go Jogging!
Edgar prefers to jog around Kawit Municipal Hall while Ramon prefers to jog at Freedom
Park. They agree that the one who will score more in simplifying radicals will decide as to
where they will go for jogging. Simplify.
Edgar‟s answer
Ramon‟s answer
1.
2.
3.
4.
5.
So, where do you think they will go for jogging? Put a check mark.
47
Photo Credit: Google Image
Kawit Municipal Hall
Freedom Park
Let’s Do More!
Choose the radicals that are already in its simplest form.
1.
2.
3.
4.
5.
6.
7.
8.
Challenge Yourself!
Simplify.
1.
2.
3.
4.
5.
Topic: Operations on Radicals Sub-topic: Addition and Subtraction of Radicals
OBJECTIVES
48
a. Differentiate like and unlike radicals
b. Perform addition and subtraction of radicals
c. Appreciate historical facts of Cavite
THINGS TO REMEMBER:
Like radicals are radicals that have the same index and the same radicand .
Unlike radicals are radicals that have different radicands or different indexes, or
both.
To combine like radicals, add or subtract their coefficients and annex their
common radical.
To combine unlike radicals, simplify each radical if possible, and then combine
like radicals.
Let’s Do This!
Perform.
1.
Add the coefficients then affix the common radical
2.
Subtract the coefficients then affix the common radical.
3.
Combine like radicals.
4.
Write the perfect square factor.
Use product property.
Simplify.
Subtract like radicals
5.
Factor each radicand.
Simplify each radicand.
Simplify.
Combine like radicals.
I Can Do This!
49
What was its name before?
1998
1976
Pulo Ni
_____________
Island Cove Resort and
Leisure Park
Covelandia
To help you unfold the history, solve the radicals in Column A and match these with
the correct
Column A
Column B sums/differences in
column B. Write the
letter that corresponds to your answer on the space provided.
Y
1._____
2.
_____
3.
_____
4.
_____
5._____
6.
_____
7.
_____
8.
_____
N
G
_____
9.
Let’s Do More!
Solve.
1. Express the perimeter of the
simplest radical form.
figure in
2. On the way to softball practice,
Maggie walks
diagonally through a square field and a rectangular field. The square field has a length of
60 yards. The rectangular field has a length of 70 yards and a width of 10 yards. What is
the total distance Maggie walks through the fields?
3. Marc said that
. Do you agree with Marc? Explain why or why not?
Challenge Yourself!
50
Combine the Radicals.
1.
2.
3.
4.
5.
Topic: Operations on Radicals Sub-topic: Multiplication of Radicals
OBJECTIVES
a. Perform multiplication on radical expressions
b. Solve problems involving multiplication of radical expressions
c. Appreciate architectural skills in accurate planning
51
THINGS TO REMEMBER:
To multiply radicals of th e same order, use the property 𝑛 𝑎𝑏 = 𝑛 𝑎 ∙ 𝑛 𝑏,
then simplify by removing the perfect nth powers from the radicand.
To multiply binomials involving radicals, use the property for the
product of two binomials ( a ± b ) ( c ± d ) = ac (ad ± bc ) ± bc, then simplify
by removing perfect nth powers from the radicand or by combining similar
radicals.
To multiply two radicals with different indices, we follow the steps
below;
a. Write each radicand in fractional exponent.
b. Change the fractional exponents to similar fractions.
c. Change each back to radical form. (The two radicals will now have the
same index.)
d. Multiply the radicals and simplify.
Let’s Do This!
Find the product.
1.
·
Multiply the coefficients and multiply the radicands.
Factor the radicand.
Simplify.
=
=
2.
__ · __ = __ = __ 3.
by
=
=
- __· __
– __
Use the distributive property =
Simplify.
.
or
or
=
Transform each radical to similar terms.
=
52
·
Thus,
=
I Can Do This!
It is a word prayer that is annually held in Kawit every December
24 before the beginning of the midnight mass, What is it?
To reveal the word, multiply the following radical expressions and simplify the result.
Put a check mark inside the box if the expression at the right shows the correct answer, otherwise
put a cross mark. Then, pick out all the letters with a check mark.
S
1.
M 2.
A
3.
L
Y
T
4.
5
6.
H
I
8.
N
9.
K
I
S
10.
11.
12.
Answer: __ __ __ __ __ __ __ __
Let’s Do More!
Emilio is an architect and you like to help him find the exact areas of some location in
the plan that he made. Show how to find:
1. The area of the triangular roof sheet for the attic if the base is
m and the height is
m.
2. The area the rectangular wall to be covered by wall tiles whose dimensions are (
+1
) m and (
+ 3 ) m.
3. The area of the square lot where Gazebo will be placed if the side is ( + 1 ) m.
Challenge Yourself!
Multiply and simplify the result.
53
1.
2.
3.
4.
5.
Topic: Radicals Sub-topic: Solving Radical Equations
OBJECTIVES
a.
b.
c.
d.
Determine radical equations
Solve a radical equation
Identify extraneous solution
Solve word problems involving radical equation
54
e. Appreciate various festivals celebrated in the province
THINGS TO REMEMBER:
Radical equations are equations containing radicals with variables in the
radicand.
To solve equations with radicals, place the radical expression on one side of
the sign of equality, and then raise both sides of the equation to the index
power to eliminate the radical and solve as usual. Test the roots, discarding
those that are extraneous.
Extraneous solution is a solution that does not satisfy the given equation.
Let’s Do This!
1.
Solve and check:
Square both sides.
Check:
The solution is ___.
2.
Solve and check:
Square both sides.
Simplify
55
Check
The solution
is ___.
3.
Solve and check:
Isolate radical expression.
Square both sides.
Expand the right side.
Factor the trinomial.
Check: If x = 3,
If
Extraneous solution:___
Only solution: ____
I Can Do This!
Grateful Cavitenyos!
To reveal how Cavitenyos show their gratefulness, solve the following radical
equation then fill up the puzzle with the letter that corresponds to the correct answer. Show
your solution.
56
R
Y
A
S
4
W
P
I
4
1
M
G
1
6
W
5
2
E
U
G
M
1
6
3
5
6
U
T
-
3
N
4
N
G
1
5
H
3
D
2
M
3
1
L
P
6
1
W
2
2
U
Let’s Do More!
Solve the following problems.
1. The square root of 5 more than twice a number is 7. Find the number.
2. The square root of twice a certain number is subtracted from the number and the result
is 4. Find the number.
3. The square root of 5 less than 6 times a certain number is divided by the number and the
quotient is 1. Find the number.
4. The square root of the product of 4 and a number is 26. Find the number.
5. The square root of 1 more than twice a certain number is 5. Find the number.
Challenge Yourself!
Solve and check.
1.
2.
3.
4.
5.
Topic: Operation on Radicals Sub-topic: Division of Radicals
OBJECTIVES
a.
b.
c.
d.
Determine the conjugate of an expression
Perform division of radicals
Solve problems involving division of radicals
Value the natural resources of our province
57
THINGS TO REMEMBER:
To divide radicals of the same order, use the property
𝑛
𝑛
𝑎
𝑏
𝑛
𝑎
then
√𝑏
rationalize the denominator.
To divide radicals of a different order, it is necessary to first change the
radicals to the same order then rationalize the denominator.
To divide radicals with a denominator consisting of at least two terms,
we follow the steps below:
a. Write each radical in fractional exponent.
b. Change the fractional exponents to similar fractions.
c. Change each back to radical form.
d. Multiply the radicals and simplify.
A Conjugate pair consist of two binomial radical expressions that have
the same numbers but only differ in the sign that connects the binomials.
Let’s Do This!
Divide and simplify.
1.
Quotient Rule for Radical
2.
Quotient Rule for Radical
58
3.
Quotient Rule for Radical
Divide and Simplify
4.
Write each radical in fractional exponent.
Change the fractional exponents to similar fractions.
Change each back to radical form.
Find the value of the each radicand.
Divide the radicals and simplify.
5.
Multiply the numerator and denominator by its conjugate.
Simplify
I Can Do This!
A Way to the Treasure!
Perform division of radicals and simplify the following expressions. Trace your path to the
treasure box by connecting the dots bearing the correct answers.
1.
2.
3.
59
4.
5.
Let’s Do More!
Solve.
Dondon is making a replica of Municipality Map for his school project whereas the
required Area A and base b in centimetres are given. Find the height h of the map, express
in simplest form.
Kawit Map
Noveleta Map
Rosario Map
1. A =
2. A =
3. A =
b=
b=
b=
Challenge Yourself!
Divide and simplify the result.
1.
2.
3.
4.
5.
60
Topic: Radicals Sub-topic: Word Problems Involving Radicals
OBJECTIVES
a. Follow a step by step procedure in solving word problems involving radicals
b. Recognize formula necessary to solve a given problem
c. Appreciate the use of radicals in real life situation
THINGS TO REMEMBER:
To solve problems involving radicals, follow these steps:
Sketch
Define variable
Find an equation
Solve the equation
Check
Let’s Do This!
Solve.
1. The time T (in seconds) taken for a pendulum of length L (in feet) to make one full swing,
back and forth, is given by the formula
. To the nearest hundredth, how long
is a pendulum which takes 2 seconds to complete one full swing?
Sketch:
Define variable:
Find an equation:
Solve the equation:
61
Check
2. Find the length of the side of a square sign whose area is 50 square feet.
Sketch:
Define variable:
Find an equation:
Solve the equation:
Check
I Can Do This!
Solve each problem by writing an equation and solving it. Find the exact answer and simplify
it.
1. The length of a rectangle is 7 meters less than twice its width, and its area is 660 m2.
What are the length and width of the rectangle?
2. Idris and Sebastian are sailing in a friend‟s sailboat. They measure the hull speed at 9
nautical miles per hour. Find the length of the sailboat‟s waterline. Round to the nearest
foot. (The formula for hull speed is h=1.34 )
Let’s Do More!
Solve.
1. The area of a triangle is 24 in2 and the height of the triangle is twice as long as the base.
What are the base and the height of the triangle?
2. The amount of time t, in seconds, that it takes a pendulum to complete a full swing is
called the period of the pendulum, It is given by t = 2
, where is the length of the
62
pendulum, in feet. The Giant Swing completes a period in about 8 second. About how
long is the pendulum‟s arm? Round to the nearest foot.
Challenge Yourself!
Solve the following problems.
1. One form of Coulomb‟s law in electricity states that d = k
. In the formula, d is the
distance between two charged bodies, and q1 and q2 are the charges carried by the bodies.
When q1 = 1. d = 5 cm, and k = 4, find q2.
2. The approximate radius of a right circular cylinder is given by the formula r=√
, when
V is the volume and h is the height. Find the radius to the nearest centimeter of a right
cylinder with a volume 150 cubic centimeters and the height of 12 centimeters.
63