Learning
Competencies
At the endof this week, you
should be able to:
Express relationship
between two variables as
a function
Determine the slopes (as
rate of change) and the
zeros of linear functions
represented in:
a. graphs
b. equations
c. tables of values
A. Identify whatkind of relation
and tell whether function or not.
1. The total cost of sugar bought is related to
the number of kilograms of sugar bought.
2. The score obtained on a test is related to
the length of time spent in studying a
subject.
3. The distance traveled by a car is related to
the speed and the time it has traveled.
4. {(yellow, orange), (red, orange), (yellow,
green), (blue, green), (red, violet)}
5. Region I (Ilocos Norte, Ilocos Sur, La Union,
8.
B. Draw anarrow diagram and identify
what type of relation. Tell whether function
or not.
1. {(-2,3), (-1,4), (0,5), (1,6), (2,7)}
2. Distance Jeepney fare
5 km. 14.00
4 km. 13.00
3 km. 13.00
6 km. 15.00
7 km. 16.00 .
A. Identify whatkind of relation
and tell whether function or not.
1. The total cost of sugar bought is related to
the number of kilograms of sugar bought.
2. The score obtained on a test is related to
the length of time spent in studying a
subject.
3. The distance traveled by a car is related to
the speed and the time it has traveled.
4. {(yellow, orange), (red, orange), (yellow,
green), (blue, green), (red, violet)}
5. Region I (Ilocos Norte, Ilocos Sur, La Union,
Function
Function
Function
Not function
Not function
12.
B. Draw anarrow diagram and identify
what type of relation. Tell whether function
or not.
14
13
15
16
5
4
3
6
7
3
4
5
6
7
-2
-1
0
1
2
FUNCTION
FUNCTION
1. {(-2,3), (-1,4), (0,5), (1,6), (2,7)}
2. Distance Jeepney fare
5 km. 14.00
4 km. 13.00
3 km. 13.00
6 km. 15.00
7 km. 16.00 .
13.
C. Complete thetable.
ORDERED PAIR
FUNCTION
OR NOT
DOMAIN RANGE
1. {(4,10), (2,8), (4, 7), (2,9)} NOT {4, 2} {1, 3, 5, 7}
2. {(1,a), (2,b), (3,c), (4,d)} FUNCTION {1, 2, 3 ,4} {a, b, c, d}
3.{(1, 3), (3, 5), (5, 7), (7, 9)} FUNCTION {10, 8, 7, 9} {3, 5, 7, 9}
4. {(2,4), (2,8), (2,16), (2,32)} NOT {2} {4, 8, 16, 32}
5. {(3, 2), (4, 2), (5, 2), (6, 2)} FUNCTION {3, 4, 5, 6} {2}
The teacher wantsto
analyze if there is really an
effect if the students spend
more time studying.
In this lesson, you should learn the concept about the
relationship between two variables.
Suppose a teacher wants to know why some students score
better than others.
The teacher might think that it is because the students spend
more time studying for their tests while other students may
be more knowledgeable than others.
16.
This scenario canbe presented using variables
and their type.
The two types of variables are commonly used
in statistics and most of the time
experimentation.
In an experiment, they used variables to
determine if one variable influence another.
Variable
is an alphabetor
term that
represents an
unknown number
or unknown value
or unknown
quantity. It is a
quantity that can be
changed or which is
not fixed according
to the
A. Dependent
Variable
is a variable
whose value
depends on
another
variable.
B. Independent
Variable
is a variable
whose value
never depend
on another
variable
In the applicationwhere the teacher wants to
know why some students score better than the
others, the teacher asks 50 students to
complete a math test.
By doing this, the teacher wants to know if the
students who scored better than others spend
more time studying for their test.
The teacher decides to analyze the effect of
having more time studying for their test based
on the performance of the 50 students.
21.
For this application,the
• the test mark (performance of the
students which can be measured)
is the dependent variable while
• the time spent in studying is the
independent variable.
22.
• Independent variable= the variable you
choose or change
• Dependent variable = the variable that
changes because of the independent variable
In a function:
• is the independent variable
• is the dependent variable
because the value of depends on the value of .
23.
Example:
Why?
Because when changes,also changes.
Try some values:
if , then
If , then
if , then
So, depends on .
Dependent
variable
Independent
variable
1. Let nbe the number of chocolate bars purchased.
Each chocolate (C) bars cost 24.00 pesos each. Identify
the dependent and independent variable.
a. What is the total cost of the chocolate (C) bars?
Equation: C = 24.00 n
In the given example:
n is the independent variable C is the dependent variable
b. How many chocolate bars will be purchased if you pay C
amount of chocolate bars?
Equation: n =
In the given example:
C is the independent variable n is the dependent variable
26.
2. The studentcouncil plans to offer refreshments at the
next council general assembly. The number of
sandwiches that are purchased will depend upon the
number of students attending the assembly.
Let:
c = be the number of students attending the general
assembly
s = the number of sandwiches purchased
Which of the following statements is true?
a. c is the independent variable and s is the dependent
variable
27.
3. How doessalary increase affect employees’
motivation to work hard?
independent variable
- salary increase
dependent variable
- employees’ motivation
28.
4. How higherdegree can lead to a higher
income?
independent variable
- higher degree
dependent variable
- higher income
29.
5. The researcherwants to know whether
eating breakfast affects the score of the
students.
independent variable
- presence or absence of
breakfast
dependent variable
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 1: Hours Worked and Total Pay
Let = number of hours worked
Let = total pay
If the pay rate is 150 per hour, then the relationship is:
This means:
If , then
if , then
So is a function of .
32.
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 2: Cost and number of items
Let = number of notebooks
Let = total cost
If one notebook costs 25 pesos, then:
If , then:
33.
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 3: Temperature conversion
Let = temperature in Celsius
Let = temperature in Fahrenheit
The function is:
So if ,
34.
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 4: Distance and speed
Let = time in hours
Let = distance traveled
If speed is 60 km/h, then:
If ,
35.
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 5: Area of a Circle
Let r = radius
Let A = area
A(r) = πr2
The area depends on the radius.
36.
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 6: Savings over weeks
Let = number of weeks
Let = total savings
If you save 200 pesos each week:
37.
EXPRESSING THE RELATIONSHIPBETWEEN
TWO VARIABLES AS A FUNCTION
Example 7: Age after some years
Let x = number of years from now
Let A = age
If a person is now 15 years old:
A(x) = 15 + x
A. Be aDetective! Write the dependent and
independent variables from the given statement
40.
B. Complete Me!Complete the table by identifying the Dependent
and Independent Variable and the function.
Given
Independent
Variable
Dependent
Variable
Function
represents the
relationship
1. A tricycle fare (F) costs 10
pesos for every kilometer
(k) traveled.
2. A student saves (S) 50
pesos each week (w).
3. Each notebook (N) costs (C)
25 pesos.
4. A worker earns (E) 120
pesos per hour (h).
5. The distance (d) traveled
depends on the time (t) if
the speed is 60 km/h.
A. Be aDetective! Write the dependent and
independent variables from the given statement
Number of
cupcakes
Number of
classmates
Earnings
Number of hours
worked
Height the ball
bounces
Height from which
the ball is dropped
Time it takes to
walk to school
Distance from
house to school
Time for the egg
to boil
Temperature of
water
Students’ test
grades
Playing Mobile
Legends
ANSWER KEY
43.
B. Complete Me!Complete the table by identifying the Dependent
and Independent Variable and the function.
Given
Independent
Variable
Dependent
Variable
Function
represents the
relationship
1. A tricycle fare (F) costs 10
pesos for every kilometer
(k) traveled.
number of
kilometers
(k)
Fare (F) F(k)=10k
2. A student saves (S) 50
pesos each week (w). weeks (w) savings (S) S = 50w
3. Each notebook (N) costs
(C) 25 pesos.
number of
notebooks
(n)
total cost
(C)
C = 25n
4. A worker earns (E) 120
pesos per hour (h).
hours
worked (h)
earnings
(E)
E = 120h
5. The distance (d) traveled
depends on the time (t) if time (t)
distance
traveled d = 60t
ANSWER KEY
44.
C. Complete Me!Complete the table by identifying the Dependent
and Independent Variable
Given Dependent
Variable
Independent
Variable
1. The amount of money earned if raffle tickets
are sold.
amount of money
earned
number of raffle
tickets sold
2. Dan is taking a quarterly examination. His
final score will depend on the number of
questions he answers correctly.
final score number of questions
answered correctly
3. In a flower farm, new caretaker was assigned.
The caretaker wants to know if the sunflower
grows taller if they are exposed to more
sunlight.
height of the
sunflower amount of sunlight
4.
distance time
5. A bakeshop earns 80 pesos for every cake number of cakes
ANSWER KEY
B. Choose Me!Tell whether the highlighted statement is dependent
or independent variable. Shade the circle that corresponds to your
answer.
1. A food blogger receives payment from an advertiser each
month. The amount of money she earns depends on how many
site visitors she receives each month.
2. Shane is making goodie bags for her son’s birthday. She buys
bottle of bubbles to be placed inside the bags. The number of
bubbles she will buy will be determined by how many kids attend
the party.
3. Sonia is hosting a party for their reunion. The number of chairs
she will need is determine by the number of friends who will attend
the reunion.
4. Tina is collecting prize baskets for a raffle. The number of prizes
donated determines the number of baskets she makes
5. Eldan is planning a picnic for his coworkers and plan to serve
chicken burger. The more of his coworkers who attend the picnic,
the more chicken burgers he will prepare
DV IV
DV IV
DV IV
DV IV
DV IV
49.
B. Choose Me!Tell whether the highlighted statement is dependent
or independent variable. Shade the circle that corresponds to your
answer.
1. A food blogger receives payment from an advertiser each
month. The amount of money she earns depends on how many
site visitors she receives each month.
2. Shane is making goodie bags for her son’s birthday. She buys
bottle of bubbles to be placed inside the bags. The number of
bubbles she will buy will be determined by how many kids attend
the party.
3. Sonia is hosting a party for their reunion. The number of chairs
she will need is determine by the number of friends who will attend
the reunion.
4. Tina is collecting prize baskets for a raffle. The number of prizes
donated determines the number of baskets she makes
5. Eldan is planning a picnic for his coworkers and plan to serve
chicken burger. The more of his coworkers who attend the picnic,
the more chicken burgers he will prepare
DV IV
DV IV
DV IV
DV IV
DV IV
ANSWER KEY
In everyday life,many situations show how one
quantity changes in relation to another.
For example, the amount of money earned
depends on the number of hours worked, and the
distance traveled depends on the time spent
moving.
These relationships can often be represented by
linear functions, which have a constant rate of
change. This constant rate of change is called the
52.
In this lesson,students will learn how to
determine the slope of linear functions
from different representations, such as
graphs, equations, and tables of values.
By doing this, they will better understand
how slope describes the change between
two variables and how it can be used to
solve real-life problems.
Try to answerthe following:
The equation y = mx + b is a linear equation where "m"
represents the (1)____________ and "b" represents the
(2)_________________. The (3)_________________ is the one
we control or change, and the (4)_______________
depends on the independent variable in a linear
relationship. The (5)_______________ is the horizontal
number line on a coordinate plane, and the
(6)________________ is the vertical number line. The
steepness of a line is called its (7)_____________________.
Slope y – intercept independent variable
x – axis y – axis dependent variable
Slope
y-intercept independent
variable
dependent
variable
x-axis
y-axis
Slope
Class, in ourprevious lesson, we
learned about relation and function.
We said that a function is a rule that
matches each input with only one
output.
Today, we will focus on one special
kind of function called a linear
function.
CHARACTERISTICS OF ALINEAR FUNCTION
• It has a constant rate of change.
If one variable increases, the other increases or
decreases by the same amount each time.
Example: S = 50w
The savings increase by 50 pesos for every 1 week.
Constant rate = 50 pesos per week
• Its equation can be written as y = mx + b.
Here, m is the constant rate, and b is the starting value.
Example: y = -3x + 5, f(x) = 2x – 6
• The variable has an exponent of 1 only.
Example: x is linear, but x² is not linear.
• Its graph is a straight line.
Example:
60.
We can recognizea linear
function through its
• graph,
• table of values, or
• equation.
61.
• In thegraph, it appears as a straight line.
• In the table, the change in values is consistent.
• In the equation, it is usually written in the form y = mx +
b or f(x) = mx + b, where m is the slope and b is the y-
intercept.
x -3 -2 -1 0 1 2
y -5 -3 -1 1 3 5
x 5 10 15 20 25 30
y -110 -120 -130 -140 -150 -160
+1
}
+1
}
+1
}
+2
}
+2
}
+2
}
+5
}
+5
}
+5
}
-10
}
-10
}
-10
}
y = 2x + 3 f(x) = -4x + 3 y = ¼ x - 5
A. Encircle the
linearfunctions.
f(x) = x - 2
f(x) = 3x2
+ 2x + 5
f(x) = -5x
f(x) = 5
f(x) = 7xy
f(x) = x + 10
f(x) = 2x4
+ 3
f(x) = 2x
- 3
f(x) = 3 + 5x
B. Linear or not linear
ANSWER KEY
LINEAR
LINEAR
NOT LINEAR
NOT LINEAR
LINEAR
• In thegraph, it appears as a straight line.
• In the table, the change in values is consistent.
• In the equation, it is usually written in the form y = mx +
b or f(x) = mx + b, where m is the slope and b is the y-
intercept.
x -3 -2 -1 0 1 2
y -5 -3 -1 1 3 5
x 5 10 15 20 25 30
y -110 -120 -130 -140 -150 -160
+1
}
+1
}
+1
}
+2
}
+2
}
+2
}
+5
}
+5
}
+5
}
-10
}
-10
}
-10
}
y = 2x + 3 f(x) = -4x + 3 y = ¼ x - 5
A linear functionalways involves slope
because slope helps us describe how the
line goes up or down.
Slope tells us how steep a line is and
shows the direction of the line. It tells us
how much the value of changes for
𝑦
every change in .
𝑥
The symbol for slope is “m”
70.
Remember this: thebigger the slope, the steeper the line.
This means that as the slope gets larger, the line goes up faster.
m = 1 m = 2
a b
Line b is steeper than line a
71.
THE SLOPE OFA LINEAR FUNCTION
m = positive
rises from left
to right
(increasing)
m = negative
falls from left
to right
(increasing)
m = 0
horizontal line
m = undefined
vertical line
THE SLOPE OFA LINEAR FUNCTION
+3
+1
GIVEN: GRAPHS
m =
rise
run
m =
+1
+3
m =
1
3
75.
THE SLOPE OFA LINEAR FUNCTION
+3
GIVEN: GRAPHS
m =
rise
run
m =
0
3
m = 0
76.
GIVEN:
Table of Values
THESLOPE OF A LINEAR FUNCTION
m = Change in y
Change in x
m = +2
+1
2
m =
x -3 -2 -1 0 1 2
y -5 -3 -1 1 3 5
+1
}
+1
}
+1
}
+2
}
+2
}
+2
}
77.
THE SLOPE OFA LINEAR FUNCTION
x 5 10 15 20 25 30
y -110 -120 -130 -140 -150 -160
+5
}
+5
}
+5
}
-10
}
-10
}
-10
}
GIVEN:
Table of Values
m = Change in y
Change in x
m = -10
+5
-2
m =
78.
THE SLOPE OFA LINEAR FUNCTION
y = -x + 6
y = mx + b
m = -1
f(x) =
f(x) = mx + b
m =
GIVEN: Equation
79.
THE SLOPE OFA LINEAR FUNCTION
4x + y = -5
y = mx + b
y = -4x – 5
m = -4
-x - 4y = -10
y = mx + b
m = -
GIVEN: Equation
The zero ofa linear function is the
value of x that makes the function
equal to zero.
In other words, it is the x-value when y
= 0.
The zero of a linear function can be
determined in different ways
88.
In a graph,the zero is found by
locating the point where the line
crosses the x-axis.
Since all points on the x-axis have
a y-value of 0, the x-coordinate of
that point is the zero of the
function.
89.
In a tableof values, the zero is
found by looking for the value of x
that corresponds to y = 0.
If the table shows that y = 0 when x
= 3, then the zero of the function is
3.
90.
In an equation,the zero is
found by replacing y or f(x)
with 0 and solving for x.
For example, in the equation y
= 2x - 4, let y = 0. Then, 0 = 2x -
4. Solving gives x = 2, so the
zero of the function is 2.
(x) 0 12 3 4 5
f(x) -4 -3 -2 -1 0 1
(x) -4 -3 -2 -1 0 1
f(x) -2 -1 0 1 2 3
(x) 0 1 2 3 4 5
f(x) -6 -4 -2 0 2 4
(x) -4 -3 -2 -1 0 1
f(x) -6 -3 0 3 6 9
B. TABLE OF
VALUES 1)
2)
3)
4)
To find the
zero of a linear
equation from
a table of
values, look
for the value of
x when y = 0.
95.
y = x+ 2
0 = x + 2
-x = 2
x = -2
Or (-2,0)
y = 4x - 2
0 = 4x - 2
4x = 2
=
x =
C. EQUATIONS
96.
2x – 3y= -6
2x – 3(0) = -6
2x = -6
=
x = or (-3,0)
C. EQUATIONS
