Table of Contents
On the coordinate plane, every point is described by an ordered pair of numbers, usually written as $(x, y)$. In this chapter, you will practice how to go from numbers to a location on the plane, and from a location on the plane back to numbers.
Remembering the axes and order
From the parent chapter, you already know:
- The horizontal axis is the $x$-axis.
- The vertical axis is the $y$-axis.
- They cross at the origin, $(0, 0)$.
Each point on the plane is written as an ordered pair:
$$
(x, y)
$$
The order matters:
- The first number is the $x$-coordinate (horizontal).
- The second number is the $y$-coordinate (vertical).
A common way to remember: “walk before you climb” or “across, then up/down.” You move horizontally first (along $x$), then vertically (along $y$).
Step-by-step: how to plot a point
To plot a point given its coordinates $(x, y)$:
- Start at the origin $(0, 0)$.
- Move horizontally:
- If $x$ is positive, move $x$ units to the right.
- If $x$ is negative, move $x$ units to the left.
- If $x = 0$, stay on the $y$-axis.
- From that new position, move vertically:
- If $y$ is positive, move $y$ units up.
- If $y$ is negative, move $y$ units down.
- If $y = 0$, stay on the $x$-axis.
- Mark the point and label it with its coordinates or a letter (for example, $A(3, -2)$).
Examples
- Plot $(3, 2)$:
- From $(0, 0)$, move $3$ units to the right (because $3$ is positive).
- From there, move $2$ units up (because $2$ is positive).
- Mark this location and label it $(3, 2)$.
- Plot $(-4, 1)$:
- From $(0, 0)$, move $4$ units to the left (because $-4$ is negative).
- From there, move $1$ unit up.
- Mark this location and label it $(-4, 1)$.
- Plot $(2, -3)$:
- From $(0, 0)$, move $2$ units to the right.
- From there, move $3$ units down (because $-3$ is negative).
- Mark this location and label it $(2, -3)$.
- Plot $(-5, -2)$:
- From $(0, 0)$, move $5$ units to the left.
- From there, move $2$ units down.
- Mark this location and label it $(-5, -2)$.
Special cases: on the axes and at the origin
Some points sit directly on the axes.
- A point on the $x$-axis has $y = 0$, for example:
- $(3, 0)$: $3$ units to the right, no vertical movement.
- $(-2, 0)$: $2$ units to the left, no vertical movement.
- A point on the $y$-axis has $x = 0$, for example:
- $(0, 4)$: no horizontal movement, $4$ units up.
- $(0, -3)$: no horizontal movement, $3$ units down.
- The origin $(0, 0)$ is where the axes cross.
- No movement from the starting point; you are already there.
Being able to quickly see whether a point lies on an axis is useful: just check whether one of the coordinates is $0$.
Using quadrants to check your plotting
The coordinate plane is divided into four quadrants. While the idea of quadrants is introduced in the parent section, here we use them as a tool to check if a point is plotted in the right general area.
- Quadrant I: $x > 0$, $y > 0$ (right and up)
- Quadrant II: $x < 0$, $y > 0$ (left and up)
- Quadrant III: $x < 0$, $y < 0$ (left and down)
- Quadrant IV: $x > 0$, $y < 0$ (right and down)
When you plot a point, check:
- If both coordinates are positive, your point should be in Quadrant I.
- If $x$ is negative and $y$ is positive, it should be in Quadrant II.
- If both are negative, Quadrant III.
- If $x$ is positive and $y$ is negative, Quadrant IV.
- If one coordinate is zero, the point is on an axis (not in any quadrant).
This is a quick way to catch mistakes, especially sign errors.
Quick quadrant checks
- $(4, 5)$:
- $x > 0$, $y > 0$ → Quadrant I.
- $(-3, 6)$:
- $x < 0$, $y > 0$ → Quadrant II.
- $(-1, -4)$:
- $x < 0$, $y < 0$ → Quadrant III.
- $(2, -7)$:
- $x > 0$, $y < 0$ → Quadrant IV.
- $(0, -2)$:
- On the $y$-axis (not in a quadrant).
Common mistakes when plotting points
Being aware of typical errors will help you avoid them:
- Reversing the order of coordinates
Confusing $(x, y)$ with $(y, x)$ changes the point.
- $(2, 5)$: right 2, up 5.
- $(5, 2)$: right 5, up 2.
These are not the same location. Always remember the order: $(x, y)$.
- Mixing up horizontal and vertical movement
- Horizontal (left–right) uses the $x$-coordinate.
- Vertical (up–down) uses the $y$-coordinate.
If you use $y$ for horizontal movement or $x$ for vertical, the point will be wrong.
- Ignoring negative signs
- A negative $x$ value means left, not right.
- A negative $y$ value means down, not up.
For example, confusing $(-3, -2)$ with $(3, 2)$ places the point in the completely opposite quadrant.
- Not starting from the origin each time
Each time you plot a new point, your mental process should start at $(0, 0)$. Do not measure from where you plotted the previous point.
Reading coordinates from a plotted point
Sometimes you are given a graph with a labeled point and asked to write its coordinates.
To find the coordinates of a point that is already plotted:
- From the point, move horizontally (left or right) to the $y$-axis and read the $x$-value.
- From the point, move vertically (up or down) to the $x$-axis and read the $y$-value.
- Write the ordered pair as $(x, y)$.
Example:
- Suppose point $A$ is plotted so that if you move horizontally from $A$ to the $y$-axis, you hit $x = -4$, and if you move vertically from $A$ down to the $x$-axis, you hit $y = 3$.
- Then $A$ has coordinates $(-4, 3)$.
Again, use quadrant knowledge to check: $x < 0$, $y > 0$ means $A$ should be in Quadrant II.
Plotting multiple points and simple shapes
You will often be asked to plot several points and then connect them.
Steps:
- Plot each point carefully using the process above.
- Label each point with its name, like $A(1, 2)$, $B(3, 2)$, and so on.
- Connect specific points with straight lines when instructed.
Example: Plot and connect $A(1, 1)$, $B(4, 1)$, $C(4, 3)$, and $D(1, 3)$ in order and return to $A$.
- Plot $A(1, 1)$, $B(4, 1)$, $C(4, 3)$, $D(1, 3)$.
- Draw line segments $AB$, $BC$, $CD$, and $DA$.
- You will see a rectangle.
Plotting points is the basis for graphing lines, curves, and more complicated figures later on.
Working with fractional and decimal coordinates
Coordinates are not always whole numbers. The process is the same; you just place the point between the whole-number marks.
- For $(1.5, 2)$:
- Move right past $1$ to halfway between $1$ and $2$.
- Then move up $2$ units.
- For $(-3, 2.5)$:
- Move left to $-3$.
- Then move up to halfway between $2$ and $3$.
When plotting fractional or decimal coordinates, try to place the point as accurately as possible between the tick marks.
Practice ideas
To get comfortable with plotting points, you can:
- Choose random pairs of integers between $-5$ and $5$ and plot them.
- For each point, say (or write):
- Which quadrant it belongs to, or whether it lies on an axis.
- Draw a simple picture (like a house or a letter) by choosing and plotting points, then connecting them.
Plotting points accurately and reading coordinates correctly are essential skills you will use later for graphing lines, studying functions, and solving many algebra and geometry problems.