Math Analysis – Precalculus, Sullivan 10th Edition
Ch1Sec1 1 8/6/2019
2 4–2–4 x
2
4
–2
–4
y
O
Section 1.1 – The Distance and Midpoint Formulas
Rectangular Coordinates
We locate a point on the real number line by assigning it a single real number, called the coordinate of the point.
For work in a two-dimensional plane, we locate points by using two numbers.
Begin with two real number lines located in the same plane: one horizontal and the other vertical. The
horizontal line is called the x-axis, the vertical line the y-axis, and the point of intersection the Origin O. Assign
coordinates to every point on these number lies using a convenient scale. In mathematics, we usually use the
same scale on each axis, but in applications, different scales appropriate to the application may be used.
The origin O has a value of 0 on both the x-axis and the y-axis. Points on the x-axis to the right of O are
associated with positive real numbers, and those to the left of O are associated with negative real numbers.
Points on the y-axis above O are associated with positive real numbers, and those below O are associated with
negative real numbers. The x-axis and y-axis are labeled as x and y, respectively, and an arrow at the end of
each axis is used to denote the positive direction.
The coordinate system described here is called a rectangular or Cartesian coordinate system. The plane
formed by the x-axis and y-axis is sometimes called the xy-plane, and the x-axis and y-axis are referred to as
the coordinate axes.
Any point P in the xy-plane can be located by using an ordered pair
)y,x(
of real numbers. Let x denote the
signed distance of P from the y-axis (signed means that if P is to the right of the y-axis, then x > 0, and if P is to
the left of the y-axis, then x < 0); and let y denote the signed distance of P from the x-axis. The ordered pair
(x, y), also called the coordinates of P, gives us enough information to locate the point P in the plane.
The origin has coordinates (0, 0). Any point on the x-axis has coordinates of the form (x, 0), and any point on
the y-axis has coordinates of the form (0, y).
If
)y,x(
are the coordinates of a point P, then x is called the x-coordinate, or abscissa, of P and y is the
y-coordinate, or ordinate, of P. We identify the point P by its coordinates (x, y) by writing P = (x, y). Usually,
we will simply say “the point (x, y)” rather than “the point whose coordinates are (x, y).”
The coordinate axes divide the xy-plane into four sections, called quadrants. In quadrant I, both the x-
coordinate and the y-coordinate of all points are positive; in quadrant II, x is negative and y is positive; in
quadrant III, both x and y are negative; and in quadrant IV, x is positive and y is negative. Points on the
coordinate axes belong to no quadrant.
y, ordinate
•
P=(x,y)
Quadrant II Quadrant I
0y,0x
0y,0x
x, abscissa
Quadrant III Quadrant IV
0y,0x
0y,0x
