An * ordered pair* of real numbers can
be represented by a point in the plane. Construct a pair of
perpendicular lines, one horizontal (the *
x-axis*), the other vertical (the * y-axis*).
Locate the point on the plane corresponding to by starting at the
* origin* --- the place where the axes cross ---
and moving x units horizontally and y units vertically. If x is
positive, you move to the right; if x is negative, you move to the
left. Likewise, if y is positive, you move up; if y is negative, you
move down.

x and y are the * Cartesian coordinates* of the
point.

* Example.* Plot the points ,
, , , and on a set of
coordinate axes.

The * midpoint* of a segment is the point halfway
between the two points. If the points are and ,
the midpoint is

* Example.* Find the midpoint of the segment
joining the points and .

The midpoint is .

The * distance* between points
and can be found using Pythagoras' theorem. It is

* Example.* Find the distance between the
points and .

An equation involving x and y can be represented by a set of points
in the plane --- the * graph* of the equation.
The graph of an equation consists of all points whose coordinates
satisfy the equation.

* Example.* Graph .

Select some x-values and plug them into the equation to find the corresponding y-values. Then plot the resulting points.

* Example.* Graph .

* Example.* Graph .

The * x-intercepts* of an equation are the places
where the graph crosses the x-axis. You can find the x-intercepts by
setting and solving for x.

The * y-intercepts* of an equation are the places
where the graph crosses the y-axis. You can find the x-intercepts by
setting and solving for y.

* Example.* Find the x-intercepts and
y-intercepts of .

To find the y-intercepts, set :

To find the x-intercepts, set :

* Example.* Find the x-intercepts and
y-intercepts of .

To find the y-intercepts, set :

To find the x-intercepts, set :

In what follows, I'll say two equations are * the
same* if you can get from either one to the other using valid
algebra.

You can often use * symmetry* as an aid in
graphing equations.

The graph of an equation is * symmetric about the
y-axis* if the equation is the same when x is replaced with .

* Example.* is symmetric
about the y-axis. If I replace x with , I get

which is the same as the original equation.

On the other hand, consider . If I replace x with , I get

This is not the same as the original equation, so the graph is not symmetric about the y-axis.

The graph of an equation is * symmetric about the
x-axis* if the equation is the same when y is replaced with .

* Example.* is symmetric about the
x-axis. If I replace y with , I get

which is the same as the original equation.

On the other hand, consider . If I replace y with , I get

This is not the same as the original equation, so the graph is not symmetric about the x-axis.

The graph of an equation is * symmetric about the
origin* if the equation is the same when x is replaced with and y
is replaced with .

* Example.* is symmetric about the origin.
If I replace x with and y with , I get

This is the same as the original equation.

On the other hand, consider . If I replace x with and y with , I get

This is not the same as the original equation, so the graph is not symmetric about the origin.

An equation of the form

represents a * circle* of radius r whose center
is the point .

* Example.* The equation

represents a circle of radius with center .

* Example.* Find an equation for the circle of
radius 3 whose center is .

* Example.* Find the radius and the center of
the circle .

I need to add a number to to make a perfect square.

so I add 9 to both sides:

Since , the radius is 8. The center is .

* Example.* Find the radius and the center of
the circle .

I need to add numbers to and to to make perfect squares.

Thus, I need to add 1 to the first expression and 25 to the second. Since I'm adding to the left side, I must add 26 to the right side of the equation as well.

The center is and the radius is 6.

Copyright 2008 by Bruce Ikenaga