∑∫ax²+bx+c=0π√x

On this page

By **MathPal Editorial Team**·PublishedApr 13, 2026·Updated Apr 14, 2026

> **Grade:** 6–7 \| **Topic:** Geometry

## What You Will Learn

By the end of this page, you will be able to identify the x-axis, y-axis, origin, and four quadrants of the coordinate plane, plot any ordered pair (x,y) accurately, and read the coordinates of a point from a graph. This skill is the foundation for graphing equations and understanding geometry.

## Theory

### The Coordinate Plane

The **coordinate plane** (also called the Cartesian plane) is a flat grid formed by two perpendicular number lines:

- **x-axis:** horizontal number line (left = negative, right = positive)
- **y-axis:** vertical number line (down = negative, up = positive)
- **Origin:** the point where the axes cross, written as (0,0)

### Ordered Pairs

Every point on the coordinate plane is described by an **ordered pair** (x,y):

- The **x-coordinate** (first number) tells you horizontal position
- The **y-coordinate** (second number) tells you vertical position

Always move horizontally (x) first, then vertically (y).

### The Four Quadrants

| Quadrant | x sign | y sign | Example |
| --- | --- | --- | --- |
| I (top-right) | + | + | (3,5) |
| II (top-left) | − | + | (−4,2) |
| III (bottom-left) | − | − | (−1,−6) |
| IV (bottom-right) | + | − | (7,−3) |

Points on the axes belong to no quadrant: (5,0) is on the x-axis, (0,−2) is on the y-axis.

## Worked Examples

### Example 1: Plotting Points in All Four Quadrants

**Problem:** Plot the points A(4,3), B(−2,5), C(−3,−4), D(5,−1).

**Point A(4, 3) — Quadrant I:**

- Start at origin. Move 4 right, then 3 up. Mark the point.

**Point B(-2, 5) — Quadrant II:**

- Move 2 left, then 5 up. Mark the point.

**Point C(-3, -4) — Quadrant III:**

- Move 3 left, then 4 down. Mark the point.

**Point D(5, -1) — Quadrant IV:**

- Move 5 right, then 1 down. Mark the point.

### Example 2: Reading Coordinates from a Graph

**Problem:** A point is located 3 units to the left of the y-axis and 7 units above the x-axis. What are its coordinates?

**Step 1:** Left of y-axis means negative x: x=−3.

**Step 2:** Above x-axis means positive y: y=7.

**Answer:** The point is (−3,7) — Quadrant II.

### Example 3: Finding Distance Along a Horizontal or Vertical Line

**Problem:** Points P(2,4) and Q(8,4) are plotted. What is the distance between them?

**Step 1:** Both points have the same y-coordinate (4), so they lie on a horizontal line.

**Step 2:** Distance = difference in x-coordinates. 
∣8−2∣=6

**Answer:** The distance is **6 units**.

### Example 4: Plotting a Shape

**Problem:** Plot the rectangle with vertices A(1,1), B(5,1), C(5,4), D(1,4). Find its area.

**Step 1:** Plot all four points and connect them in order.

**Step 2:** Width = horizontal distance from A to B = 5−1=4 units.

**Step 3:** Height = vertical distance from A to D = 4−1=3 units.

**Step 4:** Area = width × height.
A=4×3=12 square units.

**Answer:** Area = **12 square units**.

## Common Mistakes

**Mistake 1: Reversing x and y When Plotting**

❌ To plot (3,7), student moves 7 right and 3 up.

✅ The first coordinate is always x (horizontal), second is y (vertical). (3,7): move 3 right, then 7 up.

**Mistake 2: Moving in the Wrong Direction for Negative Coordinates**

❌ For (−4,2), student moves 4 right (ignoring the negative sign).

✅ Negative x means left. Move 4 left, then 2 up.

**Mistake 3: Confusing the Origin's Coordinates**

❌ The origin is at (1,1).

✅ The origin is at (0,0) — the exact crossing point of the two axes.

## Practice Problems

Try these on your own before checking the answers:

1. In which quadrant is the point (−5,3)?
2. Plot and describe the location of (0,−4).
3. What are the coordinates of a point 6 units right and 2 units below the origin?
4. Points M(−3,2) and N(−3,−5) lie on the same vertical line. What is the distance between them?
5. A square has one corner at (2,1) and the opposite corner at (6,5). What is its area?

**Click to see answers**

1. Quadrant **II** (negative x, positive y)
2. (0,−4) is on the **y-axis**, 4 units below the origin — not in any quadrant
3. (6,−2) — Quadrant IV
4. Same x-coordinate, so distance =∣2−(−5)∣=7 units
5. Side length =∣6−2∣=4 units. Area = 16 square units

## Summary

- The coordinate plane has two axes: x (horizontal) and y (vertical), crossing at the origin (0,0).
- An ordered pair (x,y): always move horizontally first (x), then vertically (y).
- Negative x = left; positive x = right; negative y = down; positive y = up.
- The four quadrants are numbered I–IV counterclockwise from the top-right.