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U4 Full Chapter 11- Graphing in the Coordinate Plane

11.1 Introduction to Graphing

A graph may be regarded as a pictorial representation of data. A graph may be drawn

manually on paper, or by using software on a graphing calculator or computer.

Graphs, especially of functions are mostly drawn on a rectangular co-ordinate plane.

A rectangular co-ordinate plane is formed by the intersection of two number lines.

One, on the East-West directions is called the horizontal axis, and the other on the

North-South directions is called the vertical axis. The horizontal and vertical axis,

make up the co-ordinate axis, and divide the co-ordinate plane into four regions

called quadrants (fig. 1).

Why a Co-ordinate Plane? Because a point on a co-ordinate plane is also identified as

the ordered pair of numbers (horizontal co-ordinate, vertical co-ordinate), and an

ordered pair of numbers interpreted as (horizontal co-ordinate, vertical co-ordinate) is

identified as a point on a co-ordinate plane. The horizontal co-ordinate is the number

on the horizontal axis, and the vertical co-ordinate is the number on the vertical axis

associated with the point on the co-ordinate plane.

0

horizontal axis

First Quadrant

Second Quadrant

QQQuadrant

Third Quadrant

QQQuadrant

Fourth Quadrant

vertical axis

fig. 1

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In (fig. 2) the points A, B, C, D are also indentified by the attached pair of

numbers. The ordered pairs (2, 3), (-5, 4), (-3.5, -6), (3,-2.5) are the points attached.

NOTE: In fig.2, the letter x is representing the horizontal axis and in such a case the

horizontal axis becomes the x-axis, and similarly the vertical axis is the y-axis. It

should be noted that the axis can be assigned any names.

Graph of Functions:

What is a function? For our purpose, it is a set of instructions and procedures which

takes a number called the INPUT NUMBER to produce another number called the

0

x

o

r

i

z

o

n

t

a

l

y

1

2

3

4

5

1

2

3

4

-1

-2

-4

-3

-5

5

-6

-6

-2

-1

-3

-4

-5

*(2,3)

*(-5,4)

*(-3.5,-6)

*(3,-2.5)

*A(3,2)

*B(0,5)

*C(4,-5)

*A(-6,0)

fig. 2

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OUTPUT NUMBER. The set of instructions and procedures are given either by

words (e.g. XX), or by algebraic equations (e.g. YY). Therefore the function is also

given by the set of ordered pair of numbers (INPUT NUMBER, Corresponding

OUTPUT NUMBER) of the function for all possible input numbers.

Function is:

And also the set of ordered pair of numbers (INPUT NUMBER, OUTPUT

NUMBER) for each input number and its corresponding output number.

The ordered pair of numbers (INPUT NUMBER, OUTPUT NUMBER) is also a

point on a co-ordinate plane whose horizontal axis is the INPUT NUMBERS and the

vertical axis is the OUTPUT NUMBERS. For such a coordinate plane the horizontal

axis is called the INPUT-axis, and the vertical axis the OUTPUT-axis.

The graph of a function is always on such a coordinate plane, and it is all the points

with coordinates (INPUT NUMBER, corresponding OUTPUT NUMBER).

How is the graph of a function constructed?

1. Choose a letter (e.g. t) or word (e.g. distance) to represent the INPUT numbers,

and a different letter (e.g. q) or (e.g. amount) to represent the OUTPUT

numbers if they are not given. From the examples, the horizontal axis is then

the t-axis and the vertical axis the q-axis.

2. Make a table (of values) with the headings INPUT (eg t), OUTPUT (eg q), and

(INPUT, OUTPUT) {eg (t, q)}. From the possible INPUT numbers choose

about 11 different numbers if the function if not linear and 4 numbers if

it is a linear function. For each INPUT number chosen find the corresponding

OUTPUT number and the ordered pair of numbers (INPUT, OUTPUT).

3. With the INPUT and OUTPUT numbers from (2) as a guide choose a scale for

the INPUT (horizontal) and OUTPUT (vertical) axis and draw both axis.

INPUT NUMBER

Instructions

and

Procedure machine

OUTPUT NUMBER

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