7.1
Rectangular Co–ordinates and Graphs:
Introduction :
we have learnt how numbers can be represented on
a number line ( it is denoted by a point which is no dimension).Line had one
dimension. Can we locate our house as a point on a line? Don’t we need to have longitude and latitude
to locate a city on a world map?
Similarly, we need to provide the row and column position
(
The word graph means ‘to paint or ‘to draw’.
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By
convention we write positive numbers on right side of zero and negative
numbers on the
left side of zero on a number line(Refer lesson 1.1) In the
graph sheet, the horizontal line is conventionally called ‘x axis’.
Thus,
the line OX represents positive
numbers and the line OX1 (X1 is also called –x) represents
negative numbers Let us
draw a perpendicular line at O to x axis and extend this line both above and
below the x axis. Again, by convention we call this vertical line as ‘y axis’. By
convention line OY represents positive numbers and the line OY1(Y1
is also called –y) represents negative numbers. The x
axis and y axis together are called ‘coordinate axes’ The
coordinate axes divide the plane into 4 parts which we call as the ‘quadrants’ named as
Quadrant I, Quadrant II, Quadrant III, Quadrant IV in the anti clock wise direction. On the graph sheet we note markings on both x axis and
y axis at equal
distances (1cm) both to the left of O and right of O on x axis, as well as
above O and below O on y axis
at equal distances with same unit of measurement (say 1cm). 1. On
OX it will be to the right of O at 1cm, 2cm, 3cm .,. 2. On
OX1 to the left of O at -1cm, -2cm, -3cm ….. 3. On
OY above O at 1cm, 2cm, 3cm, …. 4. On
OY1 below O at -1cm, -2cm, -3cm … |
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Plotting of points on a graph
sheet:
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On I
Quadrant, mark a point P (anywhere you like, above x axis and to the right of
y axis). From P draw lines parallel to both x axis and y axis. These lines have
to meet X axis and Y axis some where forming a rectangle. The
distance from O to the point where the perpendicular line from P meets x axis
is called ‘x- coordinate’ or
‘abscissa’ of
point P. The distance
from O to the point where the horizontal line from P meets y axis is called ‘y- coordinate’ or
‘ordinate ’
of point P. In the
adjoining graph x-coordinate for P is 2 units( cms)
and y-coordinate for P is 4
units(cms) and P can be represented as P(2,4), these coordinates are also called ‘rectangular
coordinates’ (because the closed figure is a rectangle) |
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7.1
Problem 1: Plot
the point P(3, 2) on the plane.
Solution:
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Step 1:
On a graph sheet mark(x1) on x axis at 3cm to the right of O Step 2:
On the same sheet mark (y1) on y axis at 2cm above O Step 3:
Construct a rectangle with sides Ox1 and Oy1 in Quadrant I. The
point where lines parallel to Ox1 and OY meet is P (3, 2) Exercise: Plot the point T(2, 3) on the plane. Do you notice
that points P (3, 2), T (2, 3) are not one and the same? Since no two
points have same coordinates x and y, the coordinate (x, y) is called ‘ordered pair’. Exercise: Mark points Q(-2,
4), R(-2, -4), S(2, -4) on the plane. By
convention note that 1. Negative x coordinate is marked to the left
of O on x axis (i.e. on line X1O) 2. Negative y coordinate is marked below the x
axis (i.e. on line OY1) We
notice that: 1.
Point Q (-2, 4) is in Quadrant II 2.
Point R (-2, -4) is in Quadrant III 3. Point
S (2, -4) is in Quadrant IV |
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Observations:
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1. The
coordinates of the origin O is (0, 0). 2. The
coordinates of any point on x axis is ( 3. The coordinates of any point on y axis is (0,
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x, 0).
7.1.1
Drawing graph for a linear equation:
We know that linear equation is an equation involving only
linear polynomial or variable in first degree.
Let us consider the below mentioned relations. In these
relations let the other number be x and the first number be y
|
No. |
Relationship |
Equivalent equation |
|
1 |
A
number is equal to other number |
y = x |
|
2 |
A
number is twice the other number |
y = 2x |
|
3 |
A
number is one more than twice the
other number |
y =
2x+1 |
|
4 |
A
number is twice (the sum of other number increased by 2) |
y = 2(x+2) |
|
5 |
Difference
between the other number and the number is 3 |
x-y =3 or
–y = 3-x(transposition) or y =
x-3(multiply by -1) |
|
6 |
Sum of
number and other number is 3 |
x+y= 3 or y = 3-x(transposition) |
All the above equations are of first degree and are of
general form y = mx
+ c where c is a constant. Even if the given equation is of the form ax+by+c = 0 it can always be converted to the form y =mx+c. How ?
|
1 |
ax+by+c = 0 |
Given. |
|
2 |
by= -ax-c |
Transposition(subtract
both sides by ax+c) |
|
3 |
y=
(-a/b)x-(c/b) |
Divide
both sides by b |
|
4 |
y =mx+z |
m=
-(a/b), z= -(c/b) |
7.1.1 Problem 1: Draw the graph for the equation x+y
=3
Solution:
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Step 1 : Convert the given equation to
the form of y = (i.e. LHS will
have only y). So we have y = 3-x(transposition). Step 2
: For few values of x (though 2 is enough) get values of y and record them in a table
like:
Step 3
: Plot the points represented by (x,y) coordinates
on a graph sheet and join them to get a straight line This line
represents the equation x+y =3(or y= -x +3) Verification: How are we sure that this line
represents the equation x+y=3? For
x=0.5 find out the y-coordinate of the point on this straight line. We notice
that it is 2.5 and thus the point (0.5, 2.5) is on the line Substituting
this value in the given equation we notice that x+y
=0.5+2.5 = 3. So any point on the straight line we have just drawn, satisfies
the given linear equation. Similarly
we find that point (3,0) is a point on the drawn
straight line and satisfies the given equation. This proves that the graph
represents the given equation. Since
y= mx+c is a first degree equation and its graph
represents a straight line, we call first degree equations as linear
equations. |
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Note :
1. Though only two points are enough to draw a straight line
we have prepared more values of (x,y)
to indicate that there are many solutions to x+y=3
2. It is also clear from the graph that many coordinates ( x, y) on the line x+y =3 {for example (4,-1)}satisfy the given equation x+y=3
Alternate Method of drawing graph:
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We know
that to draw a straight line we need only 2 points. Hence
why is it necessary to have tables of (x, y) for various values of x and y? Why not have just 2
points with one point on x axis and another point on y axis, so that by joining
these two points we can draw a straight line? A point
on x axis will always have its y coordinate as 0. Similarly a point on y axis
will always have its x coordinate as 0 Definition:
1. ‘x- intercept’
of a linear equation is the x-coordinate of
the point where graph crosses the x –axis( It is the distance of point from O
on x-axis). It’s
coordinate has to be (x,0). To get this value of x,
substitute y=0 in the given equation. 2. ‘y- intercept’
of a linear equation is the y-coordinate of
the point where graph crosses the y –axis( It is the distance of point from O
on y-axis). It’s coordinate has to be (0, y).To get this value of y,
substitute x=0 in the given equation. Note
that intercept means cut and hence x intercept means point of cutting of x
axis and y intercept means point of cutting of y axis. Let us
plot the graph using this alternate method for the equation x+y = 3(Problem 7.1.1.1) By
substituting y=0 in the above equation we get x=3. Thus P(3,0) is the
x-intercept By
substituting x=0 in the above equation we get y=3. Thus Q(0,3) is the
y-intercept By joining P and Q we get the graph for the line x+y =3.This is the same graph we have drawn in Problem
7.1.1.1 |
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7.1.1
Problem 2: Draw a
graph for y = -2
Solution:
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Step 1
: The equation can be interpreted as ‘for any value of x, y is always -2’
and hence can be represented as y =0x-2 Step 2:
For few values of x (though 2 is enough) find
values of y and record them in a table like:
Step 3 : Plot the points
represented by (x, y) coordinates on a
graph sheet and join them to get a straight line. This line represents the
equation y=-2. You can
verify that the point (2,-2) is on this drawn line also satisfies the
equation y = -2. Note that this drawn line is
parallel to x axis |
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7.1.1
Problem 3: Draw a
graph for 2y = -x.
This is of the form y=mx
Solution:
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Step 1: Convert the
given equation to the form of y = (i.e. LHS will only have y) so y = - (1/2)x Step 2: For 2 values of x find values of y and record
them in a table as shown below
Step
3: Plot the points represented by (x,
y) coordinates on a graph sheet and join them to get a straight line. This
line represents the equation y= -(1/2)x You can
verify that the point (1,-1/2) is on this drawn line and also satisfies the
equation y = -2. Note this drawn line passes through
origin (0, 0). Thus if y=mx, then the line passes
through origin. |
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7.1.1
Problem 4: Draw a
graph for x = -3. The equation can be interpreted as ‘for any value of y, x is
always -3’ or x+3=0
Solution:
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Step1: Since the equation does not have y term, we can
say 0y = -x-3
Step
2: Plot the points represented by (x,
y) coordinates on a graph sheet and join them to get a straight line. This line represents the
equation x=-3 Note that this drawn line is
parallel to y axis. |
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7.1 Summary of learning
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No |
Points to remember |
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1 |
x axis
and y axis are called co ordinate axis |
|
2 |
Any
point is represented by a ordered
pair(x, y) called coordinates of that point |
|
3 |
Origin’s
co ordinates are (0,0) |
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4 |
Any
point on the x axis has (x,0) as its coordinate and any
point on the y axis has(0,y) as its coordinate |
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5 |
The
equation to a line is of the form y = mx+c(which is
of first degree) and is called linear graph |
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6 |
The
graph x = constant is a line parallel to y axis and the
graph y = constant is a line parallel
to x axis |
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7 |
The
graph y = mx
is a line which passes through Origin |
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8 |
x-
intercept of a linear equation is the
x-coordinate of the point where
the line crosses the x –axis at (x,0) |
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9 |
y-
intercept of a linear equation is the
y-coordinate of the point where
the line crosses the y –axis at (0,y) |