Coordinate Plane - Grade 8

1. Coordinate Plane Review

Definition: A coordinate plane is a two-dimensional surface formed by two perpendicular number lines that intersect at a point called the origin.

Key Components:

1. X-Axis (Horizontal Axis):

Runs horizontally (left to right)
Positive values extend to the right
Negative values extend to the left

2. Y-Axis (Vertical Axis):

Runs vertically (up and down)
Positive values extend upward
Negative values extend downward

3. Origin:

The point where the x-axis and y-axis intersect
Has coordinates \( (0, 0) \)
The starting point for all measurements

Ordered Pairs:

Format: \( (x, y) \)

First number (x-coordinate or abscissa): Horizontal distance from the origin
Second number (y-coordinate or ordinate): Vertical distance from the origin

Plotting Points - Steps:

Start at the origin \( (0, 0) \)
Move along the x-axis (right for positive, left for negative)
From that position, move along the y-direction (up for positive, down for negative)
Mark the point

Examples:

Example 1: Plot the point \( (3, 4) \)

• Start at origin

• Move 3 units to the right

• Move 4 units up

Example 2: Plot the point \( (-2, 5) \)

• Start at origin

• Move 2 units to the left

• Move 5 units up

Important: Order matters! \( (3, 4) \neq (4, 3) \)

2. Quadrants and Axes

Definition: The x-axis and y-axis divide the coordinate plane into four regions called quadrants.

The Four Quadrants:

Quadrants are numbered I, II, III, and IV in a counterclockwise direction starting from the upper right.

Quadrant	Location	Signs	Example
Quadrant I	Upper Right	\( (+, +) \)	\( (3, 5) \)
Quadrant II	Upper Left	\( (-, +) \)	\( (-4, 2) \)
Quadrant III	Lower Left	\( (-, -) \)	\( (-2, -6) \)
Quadrant IV	Lower Right	\( (+, -) \)	\( (5, -3) \)

Detailed Descriptions:

Quadrant I: \( x > 0, y > 0 \)

Both coordinates are positive

Quadrant II: \( x < 0, y > 0 \)

x is negative, y is positive

Quadrant III: \( x < 0, y < 0 \)

Both coordinates are negative

Quadrant IV: \( x > 0, y < 0 \)

x is positive, y is negative

Points on Axes:

Points that lie directly on the axes do NOT belong to any quadrant:

On the x-axis: Form \( (x, 0) \) — Example: \( (5, 0) \), \( (-3, 0) \)
On the y-axis: Form \( (0, y) \) — Example: \( (0, 4) \), \( (0, -7) \)
At the origin: \( (0, 0) \)

Practice Questions:

Q1: In which quadrant is the point \( (-5, 3) \)?

Answer: Quadrant II (negative x, positive y)

Q2: In which quadrant is the point \( (4, -2) \)?

Answer: Quadrant IV (positive x, negative y)

Memory Tip:

"All Students Take Calculus" — Remember the quadrants going counterclockwise:

All (Quadrant I): All positive
Students (Quadrant II): Second coordinate (y) positive
Take (Quadrant III): Third quadrant (both negative)
Calculus (Quadrant IV): Cosine (x) positive

3. Follow Directions on a Coordinate Plane

Concept: Moving from one point to another by following directional instructions.

Movement Rules:

Direction	Effect on Coordinates	Example
Right	Add to x-coordinate	\( (2, 3) \) → move 4 right → \( (6, 3) \)
Left	Subtract from x-coordinate	\( (5, 2) \) → move 3 left → \( (2, 2) \)
Up	Add to y-coordinate	\( (3, 1) \) → move 5 up → \( (3, 6) \)
Down	Subtract from y-coordinate	\( (4, 7) \) → move 2 down → \( (4, 5) \)

Formula for Movement:

Starting point: \( (x_1, y_1) \)

Right/Left: New x-coordinate = \( x_1 + \text{units} \) (positive for right, negative for left)

Up/Down: New y-coordinate = \( y_1 + \text{units} \) (positive for up, negative for down)

Multi-Step Examples:

Example 1: Start at \( (2, 3) \). Move 5 units right and 2 units up. Where do you end?

Step 1: Right 5 units: \( x = 2 + 5 = 7 \)

Step 2: Up 2 units: \( y = 3 + 2 = 5 \)

Final Point: \( (7, 5) \)

Example 2: Start at \( (-1, 4) \). Move 3 units left and 6 units down. Where do you end?

Step 1: Left 3 units: \( x = -1 - 3 = -4 \)

Step 2: Down 6 units: \( y = 4 - 6 = -2 \)

Final Point: \( (-4, -2) \)

Example 3: Start at \( (5, -2) \). Move 4 units left, 3 units up, then 2 units right. Where do you end?

Step 1: Left 4: \( x = 5 - 4 = 1 \), y stays -2 → \( (1, -2) \)

Step 2: Up 3: \( y = -2 + 3 = 1 \), x stays 1 → \( (1, 1) \)

Step 3: Right 2: \( x = 1 + 2 = 3 \), y stays 1

Final Point: \( (3, 1) \)

Key Tips:

Horizontal movements only change the x-coordinate
Vertical movements only change the y-coordinate
Follow instructions in order, one at a time
Keep track of positive and negative directions

4. Find the Distance Between Two Points

Concept: The distance between two points is the length of the straight line segment connecting them.

Distance Formula:

For two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \):

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Where:

\( d \) = distance between the two points
\( (x_1, y_1) \) = coordinates of the first point
\( (x_2, y_2) \) = coordinates of the second point

Derivation (Based on Pythagorean Theorem):

The distance formula comes from the Pythagorean theorem:

• Horizontal distance: \( |x_2 - x_1| \)

• Vertical distance: \( |y_2 - y_1| \)

• Using Pythagorean theorem: \( d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \)

• Taking square root: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Steps to Calculate Distance:

Identify the coordinates of both points
Subtract the x-coordinates: \( (x_2 - x_1) \)
Subtract the y-coordinates: \( (y_2 - y_1) \)
Square both differences
Add the squared values
Take the square root of the sum

Examples:

Example 1: Find the distance between \( A(1, 2) \) and \( B(4, 6) \)

\( x_1 = 1, y_1 = 2, x_2 = 4, y_2 = 6 \)

\( d = \sqrt{(4-1)^2 + (6-2)^2} \)

\( d = \sqrt{3^2 + 4^2} \)

\( d = \sqrt{9 + 16} \)

\( d = \sqrt{25} = 5 \) units

Example 2: Find the distance between \( C(-2, 3) \) and \( D(1, 7) \)

\( x_1 = -2, y_1 = 3, x_2 = 1, y_2 = 7 \)

\( d = \sqrt{(1-(-2))^2 + (7-3)^2} \)

\( d = \sqrt{(1+2)^2 + 4^2} \)

\( d = \sqrt{3^2 + 4^2} \)

\( d = \sqrt{9 + 16} = \sqrt{25} = 5 \) units

Example 3: Find the distance between \( E(0, 0) \) and \( F(3, 4) \)

\( d = \sqrt{(3-0)^2 + (4-0)^2} \)

\( d = \sqrt{9 + 16} = \sqrt{25} = 5 \) units

Example 4: Find the distance between \( G(-3, -1) \) and \( H(2, -5) \)

\( d = \sqrt{(2-(-3))^2 + (-5-(-1))^2} \)

\( d = \sqrt{5^2 + (-4)^2} \)

\( d = \sqrt{25 + 16} = \sqrt{41} \approx 6.4 \) units

Special Cases:

1. Horizontal Line (same y-coordinates):

\( d = |x_2 - x_1| \)

Example: Distance from \( (2, 5) \) to \( (8, 5) \) = \( |8 - 2| = 6 \) units

2. Vertical Line (same x-coordinates):

\( d = |y_2 - y_1| \)

Example: Distance from \( (3, 1) \) to \( (3, 7) \) = \( |7 - 1| = 6 \) units

Key Points:

Distance is always positive (or zero if points are the same)
The order of subtraction doesn't matter (because we square the differences)
If the result is not a perfect square, leave it in radical form or use decimal approximation
Distance between a point and itself is always 0

5. Additional Formulas & Concepts

Midpoint Formula:

The midpoint is the point exactly halfway between two points:

\( M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)

Example: Find the midpoint of \( (2, 3) \) and \( (8, 7) \)

\( M = \left(\frac{2+8}{2}, \frac{3+7}{2}\right) = \left(\frac{10}{2}, \frac{10}{2}\right) = (5, 5) \)

Reflection Rules:

Over x-axis: \( (x, y) \rightarrow (x, -y) \)

Example: \( (3, 4) \rightarrow (3, -4) \)

Over y-axis: \( (x, y) \rightarrow (-x, y) \)

Example: \( (3, 4) \rightarrow (-3, 4) \)

Over origin: \( (x, y) \rightarrow (-x, -y) \)

Example: \( (3, 4) \rightarrow (-3, -4) \)

Translation:

Moving a point by adding/subtracting from coordinates:

\( (x, y) \rightarrow (x + a, y + b) \)

Example: Translate \( (2, 3) \) by moving 4 right and 2 up:

\( (2, 3) \rightarrow (2+4, 3+2) = (6, 5) \)

Quick Reference Guide

Concept	Formula/Rule
Ordered Pair	\( (x, y) \) where x = horizontal, y = vertical
Origin	\( (0, 0) \)
Quadrant I	\( (+, +) \)
Quadrant II	\( (-, +) \)
Quadrant III	\( (-, -) \)
Quadrant IV	\( (+, -) \)
Distance Formula	\( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
Midpoint Formula	\( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)