Table of Contents
The coordinate plane has already been introduced in general terms in the parent chapter. Here we focus on the specific system used on it: Cartesian coordinates.
The Cartesian coordinate system
On the coordinate plane, every point is located using a pair of numbers called its Cartesian coordinates. These numbers tell you:
- how far left or right the point is (horizontal direction),
- how far up or down the point is (vertical direction),
relative to the origin.
A point’s coordinates are written as an ordered pair:
$$(x, y)$$
- The first number, $x$, is the $x$-coordinate (or abscissa), telling you horizontal position.
- The second number, $y$, is the $y$-coordinate (or ordinate), telling you vertical position.
It is important that the order is always $(x, y)$, never $(y, x)$.
For example, the point $(3, -2)$ has:
- $x = 3$: 3 units to the right of the origin,
- $y = -2$: 2 units down from the origin.
The origin
The origin is the special point where the two axes intersect. Its coordinates are
$$(0, 0).$$
From the origin:
- Moving to the right increases $x$ (positive $x$).
- Moving to the left decreases $x$ (negative $x$).
- Moving up increases $y$ (positive $y$).
- Moving down decreases $y$ (negative $y$).
Any point’s coordinates measure signed distances from the origin along these directions.
Signs of coordinates and quadrants
The coordinate plane is divided into four quadrants by the axes. The signs of $x$ and $y$ determine which quadrant a point lies in:
- Quadrant I: $x > 0$, $y > 0$ (both coordinates positive).
- Quadrant II: $x < 0$, $y > 0$.
- Quadrant III: $x < 0$, $y < 0$ (both coordinates negative).
- Quadrant IV: $x > 0$, $y < 0$.
The quadrants are usually labeled using Roman numerals I, II, III, IV, going counterclockwise starting from the upper-right.
For example:
- $(4, 5)$ is in Quadrant I.
- $(-3, 2)$ is in Quadrant II.
- $(-1, -6)$ is in Quadrant III.
- $(5, -2)$ is in Quadrant IV.
Points on the axes are not in any quadrant.
- If $x = 0$ and $y \neq 0$, the point lies on the $y$-axis.
- If $y = 0$ and $x \neq 0$, the point lies on the $x$-axis.
- $(0, 0)$ is the origin.
Interpreting coordinates
To interpret the coordinates of a point $(x, y)$ step-by-step:
- Start at the origin $(0, 0)$.
- Move horizontally to $x$:
- right if $x > 0$,
- left if $x < 0$,
- stay on the $y$-axis if $x = 0$.
- From there, move vertically to $y$:
- up if $y > 0$,
- down if $y < 0$,
- stay on the $x$-axis if $y = 0$.
The final position is the point $(x, y)$.
For instance, for $(-4, 3)$:
- From $(0, 0)$ move 4 units left (to $x = -4$).
- Then move 3 units up (to $y = 3$).
- You arrive at $(-4, 3)$, which is in Quadrant II.
Reading coordinates from a graph
When a point is already plotted on the plane and you want to find its coordinates:
- From the point, draw a light vertical line down (or up) to the $x$-axis to read the $x$-value.
- From the point, draw a light horizontal line left (or right) to the $y$-axis to read the $y$-value.
- Write the coordinates as $(x, y)$.
Be careful:
- Count units accurately: check the scale on each axis (often 1 unit per grid square, but not always).
- Keep the sign: if the point is left of the $y$-axis, $x$ is negative; if it is below the $x$-axis, $y$ is negative.
Horizontal and vertical lines in terms of coordinates
In Cartesian coordinates, certain simple sets of points can be described by conditions on $x$ or $y$ alone:
- A vertical line through $x = a$ is all points whose $x$-coordinate is $a$:
$$x = a$$
for any $y$. - A horizontal line through $y = b$ is all points whose $y$-coordinate is $b$:
$$y = b$$
for any $x$.
For example:
- $x = 2$ is a vertical line passing through $(2, 0)$, $(2, 1)$, $(2, -3)$, etc.
- $y = -1$ is a horizontal line passing through $(0, -1)$, $(4, -1)$, $(-2, -1)$, etc.
These descriptions use the idea that each point is represented by its pair of coordinates.
Distance along one axis
In this chapter we stay with very simple distance ideas, limited to moving along a single axis.
- If two points have the same $y$-coordinate, they lie on a horizontal line. Their distance is the difference in their $x$-coordinates in absolute value:
$$\text{distance} = |x_2 - x_1|.$$ - If two points have the same $x$-coordinate, they lie on a vertical line. Their distance is the difference in their $y$-coordinates in absolute value:
$$\text{distance} = |y_2 - y_1|.$$
For example:
- Distance between $(2, 5)$ and $(-3, 5)$:
$$|2 - (-3)| = |5| = 5.$$
They are 5 units apart horizontally. - Distance between $(-4, -1)$ and $(-4, 6)$:
$$|6 - (-1)| = |7| = 7.$$
They are 7 units apart vertically.
This uses the number line ideas for absolute value, applied separately to each axis.
Using coordinates to describe position in context
Cartesian coordinates are often used to represent locations in a simple map-like setting, where each unit could represent a step, a meter, or some other distance.
For example, if a room’s floor is drawn on a coordinate grid and the origin is at one corner:
- $(3, 0)$ could be 3 units to the right along a wall.
- $(3, 4)$ could be 3 units to the right and 4 units up from the origin.
Interpreting coordinates in such contexts always follows the same rule: first move horizontally according to $x$, then vertically according to $y$.
Summary of key ideas
- A Cartesian coordinate pair $(x, y)$ pinpoints a unique location on the coordinate plane.
- $x$ gives horizontal position; $y$ gives vertical position.
- The origin is $(0, 0)$.
- Signs of $x$ and $y$ determine the quadrant or whether a point lies on an axis.
- Vertical lines have equations of the form $x = a$; horizontal lines have equations of the form $y = b$.
- Distances along a single axis are found by taking absolute differences of the corresponding coordinates.