Table of Contents
The coordinate plane is a way to locate points using pairs of numbers. You can think of it as a map made from two number lines that cross each other at right angles.
The Two Axes
In the standard coordinate plane there are two perpendicular (at $90^\circ$) number lines:
- The horizontal axis is called the $x$-axis.
- The vertical axis is called the $y$-axis.
They cross at a point called the origin, which has coordinates $(0,0)$.
Both axes are number lines:
- Numbers increase to the right on the $x$-axis and decrease to the left.
- Numbers increase upward on the $y$-axis and decrease downward.
Every point on the plane can be described using these two axes.
Ordered Pairs and Coordinates
A point in the coordinate plane is written as an ordered pair:
$$ (x, y) $$
- The first number is the $x$-coordinate (horizontal).
- The second number is the $y$-coordinate (vertical).
The order matters: $(2, 5)$ is not the same point as $(5, 2)$.
To locate $(x, y)$:
- Start at the origin $(0,0)$.
- Move $x$ units along the $x$-axis:
- Right if $x$ is positive.
- Left if $x$ is negative.
- From there, move $y$ units parallel to the $y$-axis:
- Up if $y$ is positive.
- Down if $y$ is negative.
The place where you end up is the point $(x, y)$.
Examples:
- $(3, 2)$: 3 units right, 2 units up.
- $(-4, 1)$: 4 units left, 1 unit up.
- $(2, -3)$: 2 units right, 3 units down.
- $(-1, -2)$: 1 unit left, 2 units down.
The Four Quadrants
The coordinate plane is divided into four regions called quadrants. They are numbered using Roman numerals, going counterclockwise starting from the upper right:
- Quadrant I: $x > 0$, $y > 0$ (both positive)
- Quadrant II: $x < 0$, $y > 0$ (negative $x$, positive $y$)
- Quadrant III: $x < 0$, $y < 0$ (both negative)
- Quadrant IV: $x > 0$, $y < 0$ (positive $x$, negative $y$)
Points that lie on an axis are not in any quadrant:
- If $x = 0$ and $y \neq 0$, the point is on the $y$-axis.
- If $y = 0$ and $x \neq 0$, the point is on the $x$-axis.
- If $x = 0$ and $y = 0$, the point is at the origin.
Examples:
- $(4, 5)$ is in Quadrant I.
- $(-3, 2)$ is in Quadrant II.
- $(-1, -7)$ is in Quadrant III.
- $(6, -2)$ is in Quadrant IV.
- $(0, 4)$ is on the $y$-axis.
- $(-5, 0)$ is on the $x$-axis.
Plotting Points (Sketching by Hand)
To plot a point means to draw it in the coordinate plane using its coordinates.
A practical step-by-step method:
- Draw a horizontal line and mark it as the $x$-axis with $0$ in the middle.
- Draw a vertical line through $0$ and mark it as the $y$-axis.
- Mark some positive and negative numbers on both axes at equal spacing.
- For each point $(x, y)$:
- Find $x$ on the $x$-axis.
- From that tick mark, move straight up or down to the height $y$.
- Mark the point and label it with its coordinates.
When plotting multiple points:
- Use small dots and label them clearly, for example, $A(2, 3)$, $B(-1, 4)$.
- Try to keep the scale (spacing between numbers) the same on both axes so shapes are not distorted.
Reading Coordinates from a Graph
Often you will see a point already drawn, and you must find its coordinates.
To read the coordinates:
- From the point, draw a line straight down or up to the $x$-axis to read the $x$-coordinate.
- From the point, draw a line straight left or right to the $y$-axis to read the $y$-coordinate.
- Write the ordered pair as $(x, y)$.
Check the signs:
- If the point is to the right of the $y$-axis, $x$ is positive; left means $x$ is negative.
- If the point is above the $x$-axis, $y$ is positive; below means $y$ is negative.
Distance Along Axes (Horizontal or Vertical)
In this chapter, focus on very simple distances that are purely horizontal or purely vertical.
- Horizontal distance between points with the same $y$-coordinate:
- Points $(x_1, y)$ and $(x_2, y)$ lie on a horizontal line.
- The distance is the difference in their $x$-coordinates:
$$ \text{horizontal distance} = |x_2 - x_1| $$ - Vertical distance between points with the same $x$-coordinate:
- Points $(x, y_1)$ and $(x, y_2)$ lie on a vertical line.
- The distance is the difference in their $y$-coordinates:
$$ \text{vertical distance} = |y_2 - y_1| $$
Here $|\,\cdot\,|$ means the absolute value, so the distance is always nonnegative.
Examples:
- Distance between $(2, 5)$ and $(-3, 5)$:
$$ |2 - (-3)| = |2 + 3| = 5 $$ - Distance between $(4, -1)$ and $(4, 6)$:
$$ |6 - (-1)| = |6 + 1| = 7 $$
More general distances (not purely horizontal or vertical) are treated later; for now, notice that differences in coordinates already tell you how far apart two points are along one axis.
Simple Shapes on the Coordinate Plane
Basic shapes can be drawn and described using coordinates.
Horizontal and Vertical Segments
If you connect two points:
- With the same $y$-coordinate, you get a horizontal line segment.
- With the same $x$-coordinate, you get a vertical line segment.
You can describe these segments using endpoints:
- Segment from $(1, 2)$ to $(5, 2)$ is horizontal.
- Segment from $(-3, -1)$ to $(-3, 4)$ is vertical.
Rectangles with Axis-Aligned Sides
A rectangle whose sides are parallel to the axes (an axis-aligned rectangle) has very simple coordinates.
Suppose its corners are:
- $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$ and $y_1 \neq y_2$.
Then the four vertices are:
- $(x_1, y_1)$
- $(x_1, y_2)$
- $(x_2, y_1)$
- $(x_2, y_2)$
The side lengths are:
- Horizontal side length: $|x_2 - x_1|$
- Vertical side length: $|y_2 - y_1|$
You can already see how coordinates help you describe and measure shapes.
Simple Symmetry in the Coordinate Plane
The coordinate plane helps you see symmetry using coordinates.
The most basic symmetries here are reflections across the axes:
- Reflection across the $x$-axis:
- $(x, y)$ goes to $(x, -y)$.
- Reflection across the $y$-axis:
- $(x, y)$ goes to $(-x, y)$.
Notice:
- The reflected point has the same distance from the axis as the original point, but on the opposite side.
- Only the sign of one coordinate changes.
Examples:
- Reflect $(3, 4)$ across the $x$-axis: $(3, -4)$.
- Reflect $(3, 4)$ across the $y$-axis: $(-3, 4)$.
- Reflect $(-2, -5)$ across the $x$-axis: $(-2, 5)$.
- Reflect $(-2, -5)$ across the $y$-axis: $(2, -5)$.
Understanding how coordinates change under these reflections will be helpful later when graphing more complex objects.
Using the Coordinate Plane to Organize Information
Besides geometry, the coordinate plane can be used to organize data as pairs of numbers:
- Each point can show a relationship between two quantities, such as:
- Time and distance
- Hours worked and money earned
- Temperature and day number
In these situations it is common to:
- Use the horizontal axis for the independent quantity (like time).
- Use the vertical axis for the dependent quantity (like distance).
You plot each data pair as a point $(x, y)$. Patterns in the points (such as a roughly straight line) will become important in later chapters.