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Coordinate Planes and Graphs

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Calculus and Analytical Geometry
MTH101
LECTUER ­ 3
Coordinate Planes and Graphs
Ordered pair
In this lecture we shall discuss:
By an ordered pair of real numbers we
mean two real numbers in an assigned
order. Every point P in a coordinate plane
can be associated with a unique ordered
pair of real numbers by drawing two lines
through P, one perpendicular to the x-axis
and the other to the y-axis.
A rectangular coordinate system is
a pair of perpendicular coordinate lines,
called coordinate axes, which are placed
So that they intersect at their origins.
For example if we take (a,b)=(4,3), then
on coordinate plane
The labeling of axes with letters x and y
is a common convention, but any letters may
be used. If the letter x and y are used to label
the coordinate axes, then the resulting plane
is called xy-plane. In applications it is common
to use letters other than x and y is shown In
the following figures, as uv-plane and ts-plane.
To plot a point P(a,b) means to locate
the point with coordinates (a,b) in a coordinate
plane. For example, different points are plotted.
In a rectangular coordinate system the
coordinate axes divide the plane into four
regions called quadrants. These are
numbered counterclockwise with roman
numerals as shown
Mth101
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Calculus and Analytical Geometry
Coordinate Planes and Graphs
This is an approximation to the graph of
Y=x2
. N general, it is only with techniques
from calculus that the true shape of a graph
Ca can be ascertained.
Consider the equations
We define a solution of such an equation
to be an ordered pair of
real numbers(a,b) so that the equation
is satisfactory when we substitute x=a and y=b.
Example
Sketch the graph of
Exampl
:
e
Because 1/x is undefined when x=0,
Example:  Sketch the graph of
we
can plot only points for which x=0
Mth101
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Calculus and Analytical Geometry
Coordinate Planes and Graphs
Example:
Find all intercepts of
Solution:
· symmetric about the x-axis if for each
point (x,y) on the graph the point (x,-y)
is also on the graph.
· symmetric about the y-axis if for each
point (x,y) on the graph the point (-x,y)
is also on the graph.
· symmetric about the origin, if for
is the required x-intercept.
each point (x,y) on the graph the point
(-x,-y) is also on the graph.
is the required y-intercept.
Similarly you can solve part (b), the
part (c) is solved here
In the following figure, the points (x,y),
(-x,y),(x,-y) and (-x,-y) form the corners
of a rectangle.
Mth101
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Calculus and Analytical Geometry
Coordinate Planes and Graphs
Example:
yields
Example:
Mth101
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