03/09/16
Module 3: Relations and Functions
A relation is simply
a set of ordered pairs. The first elements in the ordered pairs (the x-values),
form the domain. The second elements in
the ordered pairs (the y-values), form the range. Only the elements "used" by the
relation constitute the range.
A function is a set
of ordered pairs in which each x-element has only ONE y-element associated with
it. A function may not have two y-values assigned to the same x-value. A
function may, however, have two x-values assigned to the same y-value.
03/23/16
Activity 1: Recalling Sets
A U B
|
Red, Blue, Orange, Violet,
White
|
A ∩ B
|
Red
|
A ∩ B ∩ C
|
Red, Blue, Orange, Violet,
White, Black
|
n(A U B)
|
5
|
n(A U B)
|
1
|
A ∩ B ∩ C
|
Red, Blue
|
A ∩ B
|
Blue
|
Activity 2: The Number Line
·
The
number line is composed of numbers ranging from zero to negative or positive infinity.
·
Zero is
always the beginning of the number line.
·
Place in
number zero
o
Positive:
Left
o
Negative:
Right
Activity 3: IRF
Rectangular Coordinate: A plane that uses the
xy system.
RCS System: xy point system.
Rectangular Coordinate System
A Cartesian
coordinate system specifies each point uniquely in a plane by a pair of
numerical coordinates, are the signed distances to the point from two fixed
perpendicular directed lines, measured in the same unit of length. Each
reference line is called a coordinate axis or just axis of the system, and the
point where they meet is its origin, usually at ordered pair (0, 0). The
coordinates can also be defined as the positions of the perpendicular
projections of the point onto the two axes, expressed as signed distances from
the origin.
04/13/16
Activity 4: Locate your classmate!
Locate your seat and
the seats of group mates in the classroom. Complete the table below:
X = 4, Y = 2
|
X = 3, Y = 3
|
X = 5, Y = 2
|
Activity 5: Meet me at Thirdy’s residence
Finding a particular
point such (1, 4) in the coordinate plane is similar to finding a particular
place on the map. In this activity, you will learn how to plot points on the
Cartesian plane. With the figure at the right above, find the following
locations and label each with letters as indicated.
·
Using
the ordered pair, the axis coordinates can be used for point location.
·
An
ordered pair coordinates uses the abscissa(x) and the ordinate(y).
·
An
ordered pair is fixed to xy, unlike the counterpart yx which is just a simple
coordinate.
·
Represent the values
o
x = 3, y
= 1
§
Mabini 3rd
street, Aurora 3rd street.
o
x = 4, y
= 5
§
Mabini 5th
street, Aurora 4th street.
o
x = 1, y
= 2
§
Mabini 1st
street, Aurora 2nd street.
o
x = 4, y
= 2
§
Mabini 5rd
street, Aurora 2nd street.
Activity 6: Human Rectangular System
Form two lines. 15
of you will form horizontal line (x-axis) and 14 for the vertical line
(y-axis). These lines should intersect at the middle. Others may stay at any
quadrant separated by the lines. You may sit down and will only stand when the
coordinates of the point, the axis or the quadrant you belong is called.
Activity 7: Parts of Building
Describe the
location of each point below by completing the following table. An example is
done for. Note that the point indicates the center of the given part of the
building.
Gilt Room
|
X = +5, Y = +12
|
Quadrant I
|
Terrace Hall
|
X = -3, Y = +12
|
Quadrant II
|
Old Kitchen
|
X = -6, Y = +12
|
Quadrant II
|
Billboard Room
|
X = +8, Y = +12
|
Quadrant I
|
Salon
|
X = +2, Y = +6
|
Quadrant I
|
Reception Hall
|
X = -4, Y = -12
|
Quadrant III
|
Grand Staircase
|
X = +1, Y = -2
|
Quadrant IV
|
Marble Hall
|
X = +2, Y = -6
|
Quadrant IV
|
Reception Office
|
X = -11, Y = -5
|
Quadrant III
|
Drawing Room
|
X = +9, Y = +2
|
Quadrant I
|
Entrance
|
X = -13, Y = -2
|
Quadrant III
|
Library
|
X = +7, Y = -7
|
Quadrant IV
|
Spa
|
X = -7, Y = +7
|
Quadrant II
|
Harborough Room
|
X = +7, Y = +6
|
Quadrant I
|
Activity 8: Object’s Position
Description: This activity will enable you to give the
coordinates of the point where the object is located.
1. spoon
|
X = -5, Y = +6
|
Quadrant IV
|
2. television set
|
X = +6, Y = -5
|
Quadrant II
|
3. laptop
|
X = +2, Y = -4
|
Quadrant IV
|
4. bag
|
X = -4, Y = -3
|
Quadrant III
|
5. pillow
|
X = +5, Y = +2
|
Quadrant I
|
6. camera
|
X = +1, Y = -1
|
Quadrant II
|
7. table
|
X = +2, Y = -3
|
Quadrant II
|
Activity 9: IRF Revisit
The Rectangular
Coordinate System explains in locating objects through points and lines. The
abscissa and ordinate are the ordered pair (XY) used to locate specific points
in specific lines.
Activity 10: Spotting Erroneous Coordinates
This activity will
enable you to correct erroneous coordinates of the point.
·
Susan
indicated that A has coordinates (Y = 2, X = 4).
·
Do you
agree with Susan?
o
No
·
What
makes Susan wrong?
o
The
ordered pair is always used in plotting coordinates. XY is actually different
from its inverse counterparts. The Y is the ordinate (cuts vertically) while X
is the Abscissa (Cuts horizontally).
·
How will
you explain to her that she is wrong in a subtle way?
o
The X
should be first because it cuts through the plane horizontally.
o
The Y
should be second because it cuts through the plane vertically.
04/20/16
Representations of Relations and Functions
Activity 1: Classify!
Group the following
objects in such a way that they have common property/characteristics.
Kitchen Utensils
|
School Supplies
|
Gadgets
|
Fork
|
Notebook
|
iPod
|
Knife
|
Pencil
|
Laptop
|
Cheese Grater
|
Paper
|
Tablet
|
Ladle
|
Eraser
|
Cellphone
|
Pot
|
Pen
|
Camera
|
Activity 2: Representing a Relation
Describe the mapping
diagram below by writing the set of ordered pairs.
·
Set of
Ordered Pairs
o
Narra,
tree
o
Mohogany,
tree
o
Apricot,
tree
o
Tulip,
flower
o
Rose,
flower
o
Orchid,
flower
Activity 3: IRF
Initial Answer
|
|
Relation
|
A set of ordered pairs
|
Function
|
A set of ordered pairs with
each x-element is associated with a y-element
|
Range
|
A set containing function
output.
|
Domain
|
A set containing function
input.
|
A relation is any set of ordered pairs. The set of all first coordinates is called the domain of the relation.
The set of all second
coordinates is called the range of the relation
Activity 4: Make your own Relation
·
Exercise 1
o
Suppose
the bicycle rental at the Rizal Park is worth Php 20 per hour. Your sister would
like to rent a bicycle for amusement.
o
How much
will your sister have to pay if she would like to rent a bicycle for
§
1 hour?
2 hours? 3 hours?
·
20/1h,
40/2h, 60/3h
o
Based on
your answers in item 1, write ordered pairs in the form (time, amount).
§
1 hour,
Php 20
§
2 hour,
Php 40
§
3 hour,
Php 60
o
Based on
your answers in item 2, what is the domain? What is the range?
o
How are
rental time and cost of rental related to each other?
·
Exercise 2
o
Suppose
you want to call your mother by phone. The charge of a pay phone call is Php 5 for
the first 3 minutes and an additional charge of Php 2 for every additional
minute or a fraction of it.
o
How much
will you pay if you have called your mother in 1 minute? 2 minutes? 3 minutes?
4 minutes? 5 minutes?
§
Php 2/1
min
§
Php 4/2 min
§
Php 5/3 min
o
Out of
your answers in item 1, write ordered pairs in the form (time, charge).
§
3,5
§
4,7
§
5,9
o
Based on
your answers in item 2, what is the domain? What is the range?
§
The
domain is {3, 4, 5} while the range is {5, 7, 9}.
o
How are
time and charge related to each other?
§
The
charge of the pay phone depends on the number of minutes calling.
·
Exercise 3
o
John
pays an amount Php 12 per hour for using the internet. During Saturdays and Sundays,
he enjoys and spends most of his time playing a game especially if he is with
his friends online. He plays the game almost 4 hours.
o
How much
will John pay for using the internet for 1 hour? 2 hours? 3 hours? 4 hours?
§
Php 12/1h
§
Php 24/2h
§
Php 36/3h
§
Php 48/4h
o
Express
each as an ordered pair.
§
1, 12
§
2, 24
§
3, 36
§
4, 48
o
Is it a
relation? Explain.
§
The
amount John will have to pay depends on the time he played. The amount is 12
times the length of time
o
Based on
your answers in item 3, what is the domain? What is the range?
§
Hours –
domain
§
Cost -
range
o
How are
time and amount related to each other?
§
The
cost/amount depends on the time that passed by.
o
If John
has decided not to play the game in the internet cafe this weekend, what is the
maximum amount that he would have saved?
§
He would
have saved 48 php
05/18/16
Activity
5: Plot it
·
Determine each set of Ordered
Pairs.
·
If the x-coordinate intersects the
vertical line test, then the coordinate is a non-function.
·
If the x-coordinate does not
intersect the vertical line test, then the coordinate is a function.
o {(4,
0), (4, 1), (4, 2)}
§ Not
function
o {(0,
-2), (1, 1), (3, 7), (2, 4)}
§ Function
o {(-2,
2), (-1, 1), (0, 0), (1, 1)}
§ Function
o {(-2,
8), (-1, 2), (0, 0), (1, 2), (2, 8)}
§ Function
o {(3,
3), (0, 0), (-3, 3)}
§ Function
o {(-2,
0), (-1, √3), (-1, -√3 ), (0, 2), (0, -2), (1, √3), (1, -√3), (2, 0)}
§ Not
function
Horizontal
and Vertical Lines
|
The
horizontal line represents a function. It can be described by the equation y
= c, where c is any constant. It is called a Constant Function. However, a
vertical line which can be described by the equation x = c is not a function.
A
relation may also be represented by an equation in two variables or the
so-called rule.
|
Activity
6: Identify Me
·
Determine whether the rules of
functions apply or not.
Equation
|
Solution
|
Coordinates
|
||
y =
2x + 1
|
x = -2
y = 2x + 1 = 2(-2) + 1 = -4 +
1 = -3
|
(-2, -3)
|
||
x = -1
y = 2x + 1 = 2(-1) + 1 = -2 +
1 = -1
|
(-1, -1)
|
|||
Function
|
||||
Equation
|
Solution
|
Coordinates
|
||
x = y^2
|
(0, 0)
|
|||
(1, 1), (1, -1)
|
||||
(4, 2), (4, -2)
|
||||
Not Function
|
||||
Activity
7: Minds on
·
Dependent and Independent Variables
o The
variable x is considered the independent variable because any value could be
assigned to it. However, the variable y is the dependent variable because its
value depends on the value of x.
Function Notation
The F(x) notation
can also be used to define a function. If f is a function, the symbol F(x),
read as “F of x,” is used to denote the value of the function f at a given
value of x. In simpler way, f(x) denotes the y-value (element of the range)
that the function F associates with x-value (element of the domain).
Thus, F(1) denotes
the value of y at x = 1. Note that F(1) does not mean f times 1. The letters
such as g, h and the like can also denote functions.
Furthermore, every
element x in the domain of the function is called the pre-image. However, every
element y or F(x) in the range is called the image. The figure at the right
illustrates concretely the input (the value of x) and the output (the value of
y or F(x)) in the rule or function. It shows that for every value of x there
corresponds one and only one value of y.
06/01/16
Domain
and Range of a Function
Functions are a
correspondence between two sets, called the domain and the range. When defining
a function, you usually state what kind of numbers the domain (x) and range
(f(x)) values can be. But even if you say they are real numbers, that doesn't
mean that all real numbers can be used for x.
Activity 10: GRAPH ANALYSIS
·
By
Vertical Line Test, every graph above represents a function.
·
The
domains of the graphs are as follows:
o
First
graph: {x|x ∈ ℜ, x ≠ 0}
o
Second
graph: {x|x ≥ 0}
o
Third
graph: {x|x ∈ ℜ}
·
The
first graph does not touch the y-axis because the value of the function f
defined by f(x) = 1 x , when x = 0, is undefined, which appears Error or Math
Error in the calculator. This means that the graph of the function does not
intersect the line x = 0 or the y-axis. Thus, the domain of the function is
{x|x ∈ ℜ, x ≠ 0}.
·
In f(x)
= √x , the value of the function is a real number for every real number x which
is greater than or equal to zero. When x is negative, the value of the function
is imaginary in which calculators yield an Error or Math Error. This also means
that the graph of the function does not lies on the left side of the line x = 0
or the y-axis. Thus, the domain of the function is {x|x ≥ 0}.
·
In f(x)
= x2, there is no value of x that makes the function f undefined. Thus, the
domain of the function is {x|x ∈
ℜ}.
·
The
value of the function is not a real number when it is undefined or is imaginary.
Activity 11: IRF Worksheet Revisted
·
Relation
o
A
relation between two sets is a collection of ordered pairs containing one
object from each set. If the object x is from the first set and the object y is
from the second set, then the objects are said to be related if the ordered
pair (x, y) is in the relation. A function is a type of relation.
·
Function
o
In
mathematics, a function is a relation between a set of inputs and a set of
permissible outputs with the property that each input is related to exactly one
output. An example is the function that relates each real number x to its
square x^2.
06.22.16
Linear Function and Its Applications
Activity 1: Find my Pair
Match the verbal
phrases with their equivalent equation.
·
The sum
of the numbers x and y
o
x + y
·
The
square of the sum of x and y
o
(x + y)^2
·
The sum
of the squares of x and y
o
x^2
+ y^2
·
Nine
less than the sum of x and y
o
(x + y) –
9
·
Nine
less the sum of x and y
o
9 – (x +
y)
·
Twice
the sum of x and y
o
2(x + y)
·
Thrice
the product of x and y
o
3xy
·
Thrice
the quotient of x and y
o
3 x/y
·
The
difference between x and y divided by four
o
(x-y)/4
·
Eight
more than the product of x and y
o
8 + xy
·
The
product of 7, x and y
o
7xy
·
The
product of four and the sum of x and y
o
4(x + y)
·
The sum
of x and the square of y diminished by ten
o
x + y^2
– 10
·
Four
times the sum of the cubes of x and y
o
4(x^3
+ y^3)
·
Two
multiplied by the absolute value of the difference of x and y
o
2 |x – y|
07/28/16
Activity 2: Write Your Verbal Phrase
Direction: Write the verbal phrase for each
mathematical phrase below.
a
+ b
|
The sum of “A” and “B”
|
2(a
– b)
|
the sum of thrice the number “A” and four times the number “B”
|
3a
+ 4b
|
the sum of thrice the number “A” and four times the number “B”
|
b
– 5
|
“B”
less 5
|
5
– b
|
“B” less than 5
|
The square of the number “A” added to the square of the number
“B”
|
|
the number “A” added to twice the number “B”
|
|
Twice the square of the number
“A” diminished by thrice of the
number “B”.
|
|
The quotient of “A” and “B” added to 7.
|
Mathematical Statements
A mathematical
verbal expression is a translation into words of an algebraic expression that
can consist of different operations, numbers and variables. An example of this
is translating the mathematical equation or phrase "90 - 4(a + 8)" to
the verbal expression "90 decreased by 4 times the sum of a number
"a" and 8."
08.03.16
Activity 3: Write the Correct Equation
Represent
each of the following algebraically.
1. Twice a number is 6.
a. 2x = 6
2. Four added to a number gives ten.
a. 4 + x = 10
3. Twenty-five decreased by twice a number is
twelve.
a. 25 – 2x = 12
4. If thrice a number is added to seven, the sum
is ninety-eight.
a. 3x + 7 = 98
5. The sum of the squares of a number x and 3
yields 25.
a. x^2 + 32 = 16
6. The difference between thrice a number and
nine is 100.
a. 3x – 9 = 100
7. The sum of two consecutive integers is equal
to 25.
a. x + (x + 1) = 25
8. The product of two consecutive integers is
182.
a. x(x + 1) = 182
9. The area of the rectangle whose length is (x
+ 4) and width is (x – 3) is 30.
a. (x + 4)(x – 3) = 30
10.The sum of the ages of Mark and Sheila equals 47.
a. x + y = 47
b. M + S = 47
Activity 4: Evaluate Me
Evaluate Algebraic Expressions
2xy
|
when x = 2 and y = 1
|
|
x – 4y
|
when x =-1 and y = 0
|
|
x2 + y
|
when x = -5 and y = 7
|
|
3(x + y) – 2(x – 8y)
|
when x = 8 and y = -2
|
|
(x + 3) ÷ 4 – 15 ÷ 2xy
|
when x = 5 and y = -1
|
In mathematics, an
algebraic expression is an expression built up from integer constants,
variables, and the algebraic operations (addition, subtraction, multiplication,
division and exponentiation by an exponent that is a rational number).
Activity 5: IRF
Worksheet
·
What is a
linear function?
o Linear functions are those whose graph is a straight line. A
linear function has the following form. y = f(x) = a + bx. A linear function
has one independent variable and one dependent variable. The independent
variable is x and the dependent variable is y.
o
Total = rate x change + Fixed number
·
How do
you describe linear functions?
o
Linear function is nothing more than a constant
rate of change.
·
How do
you find the equation of a line?
o
Function = slope (value) + y-intercept
09/07/16
Activity 6: Describe Me
Think activity will
you describe linear Functions
Function
|
F(3)
|
F(1)
|
F[4]
|
F(x)= 2x
|
-6
|
2
|
4
|
F(x)= 2x+1
|
-5
|
3
|
9
|
F(x)= 3x
|
9
|
-3
|
-12
|
F(x)= 3x-4
|
5
|
7
|
16
|
Activity 7: Describe Me 2
Map each linear
function
F(x) = x + 5
|
x = -2, -1, 0, 1, 2
|
x = 3, 4, 5, 6, 7
|
F(x) = 3x
|
x = -2, -1, 0, 1, 2
|
x = -6, -3, 0, 3, 6
|
F(x) = -x + 5
|
x = -2, -1, 0, 1, 2
|
x = 7, 6, 5, 4, 3
|
F(x) = -3x
|
x = -2, -1, 0, 1, 2
|
x = 6, 3, 0, -3, -6
|
Activity 8: What are the first differences of on y values?
·
F(x) =
3x – 1
o
0 – 1 –
2 – 3 – 4
o
(-1) – 5
– 7 – 9
·
F(x) =
2x + 4
o
(-2),
(-1), 0, 1, 2
10.05.16
Slope of a Line
·
The
Mayon Volcano of the Philippines is widely known for its almost symmetrical
conical shape. To the find the steepness of the volcano, the slope of the line has
to be calculated.
·
The
slope of a line refers to the rise over run.
o
M = Rise/Run
= Vertical Change/ Horizontal Change
·
The
equation for the slope of a line is:
o
P1(x1y1)
P2(x2y2)
y2-y1
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