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L10 Rectangular Coordinates; Graphs of Equations
The Rectangular Coordinate System is formed by the
x −axis and y −axis.
A plane with the rectangular coordinate system is called the
coordinate plane or xy - plane.
This plane has four quadrants:
For each point P in the xy - plane , there is a
corresponding ordered pair ( , x y ), where x and y are
called coordinates of P , and we write P = ( , x y ).
Example: Determine the quadrant(s) in which ( , x y ) is
located if
x < 0 and y < 0
x > 2 and y = 3
Distance Formula
The distance between two points P 1 (^) = ( x 1 (^) , y 1 ) and P 2 (^) = ( x 2 (^) , y 2 ), denoted by d P P ( 1 , 2 ), is 2 2 d P P ( 1 , 2 (^) ) = ( x 2 (^) − x 1 (^) ) + ( y 2 (^) − y 1 ).
Note: 1) d P P ( 1 , 2 ) ≥ 0
- d P P ( 1 , 2 ) = 0 if and only if P 1 (^) = P 2
- d P P ( 1 , 2 (^) ) = d P ( 2 (^) , P 1 )
Midpoint
A point M on the line segment with endpoints P 1 and P 2
is called the midpoint if d P M ( 1 , ) = d M P ( , 2 ).
Graphs of Equations
An equation in two variables x and y is a statement in which two expressions involving the variables are equal.
The graph of an equation in two variables is the set of all
points ( , x y ) in the xy - plane whose coordinates satisfy
the equation.
Example. Graph the equations by plotting points. y = 2 x + 1
y = x^2
x = y^2
Intercepts
The points, if any, at which the graph crosses or touches the coordinate axes, are called intercepts.
An x-intercept is the x -coordinate of a point at which the graph crosses or touches the x -axis.
To find, we set ________ in the equation and solve for x.
A y-intercept is the y -coordinate of a point at which the graph crosses or touches the y -axis.
To find, we set ________ in the equation and solve for y.
Symmetry
The graph of an equation is symmetric with respect to the x-axis (coincides with itself when reflecting across the x -axis) if, for every point ( , x y ) on the graph, the point ( , x − y ) is also on the graph.
Test: Replacing y with − y in the equation yields in an
equivalent equation.
Example: x = y^2
The graph of an equation is symmetric with respect to the y-axis (coincides with itself when reflecting across the y -axis) if, for every point ( , x y ) on the graph, the point ( − x , y ) is also on the graph.
Test: Replacing x with − x in the equation yields in an equivalent equation.
Example: y = x^2
The graph of an equation is symmetric with respect to the origin (coincides with itself when rotated 180 D^ about the origin), if for every point ( , x y ) on the graph, the point ( − x , − y ) is also on the graph.
Test: Replacing both x with − x and y with − y in the
equation yields in an equivalent equation.
Note: Obviously, the graph is symmetric with respect to the origin if it coincides with itself when, first, reflected across the y -axis, then, across the x -axis, or vice versa.
Example: y = x^3
Example: Test the equation for symmetry.
x^3 − 3 y^2 + 1 = 0
