Coordinate Plane - Seventh Grade
Quadrants, Axes, Ordered Pairs & Distance Formula
1. Coordinate Plane Review
Definition
A coordinate plane (Cartesian plane) is a
two-dimensional surface formed by two
perpendicular number lines
• Used to locate points using ordered pairs (x, y)
• Named after French mathematician René Descartes
Parts of the Coordinate Plane
X-axis (horizontal): The horizontal number line
• Positive to the right, negative to the left
Y-axis (vertical): The vertical number line
• Positive upward, negative downward
Origin: Point where axes intersect
• Coordinates: (0, 0)
Ordered Pairs
(x, y)
x-coordinate (abscissa): First number - horizontal position
y-coordinate (ordinate): Second number - vertical position
How to Plot Points
Step 1: Start at the origin (0, 0)
Step 2: Move horizontally according to x-coordinate
• Positive x: move right
• Negative x: move left
Step 3: Move vertically according to y-coordinate
• Positive y: move up
• Negative y: move down
Step 4: Mark the point
2. Quadrants and Axes
Four Quadrants
The axes divide the plane into 4 regions called QUADRANTS
• Numbered with Roman numerals: I, II, III, IV
• Counted counter-clockwise starting from top-right
| Quadrant | Position | Signs (x, y) | Example |
|---|---|---|---|
| Quadrant I | Top Right | (+, +) | (3, 5) |
| Quadrant II | Top Left | (−, +) | (-4, 6) |
| Quadrant III | Bottom Left | (−, −) | (-2, -3) |
| Quadrant IV | Bottom Right | (+, −) | (5, -2) |
Points on Axes
On the x-axis: y-coordinate is 0
Example: (3, 0), (-5, 0)
On the y-axis: x-coordinate is 0
Example: (0, 4), (0, -7)
Important: Points on axes are NOT in any quadrant
3. Following Directions on Coordinate Plane
Movement Rules
Moving from one point to another:
Right: Add to x-coordinate
Left: Subtract from x-coordinate
Up: Add to y-coordinate
Down: Subtract from y-coordinate
Example Problem
Problem: Start at point (2, 3). Move 4 units right and 5 units down. What is the new point?
Starting point: (2, 3)
Step 1: Move 4 units right
x-coordinate: 2 + 4 = 6
Step 2: Move 5 units down
y-coordinate: 3 - 5 = -2
New point: (6, -2)
Translation Formula
Starting point: (x₁, y₁)
Move: h units horizontally, v units vertically
New point: (x₁ + h, y₁ + v)
where h is positive (right) or negative (left)
and v is positive (up) or negative (down)
4. Distance Between Two Points
Distance Formula
To find the distance between two points:
Point A: (x₁, y₁) and Point B: (x₂, y₂)
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
This is the Euclidean Distance Formula
Based on the Pythagorean Theorem
Steps to Calculate Distance
Step 1: Identify coordinates (x₁, y₁) and (x₂, y₂)
Step 2: Find difference in x-coordinates: (x₂ - x₁)
Step 3: Find difference in y-coordinates: (y₂ - y₁)
Step 4: Square both differences
Step 5: Add the squared values
Step 6: Take the square root
Example 1: Find distance between (2, 3) and (6, 7)
Given: (x₁, y₁) = (2, 3) and (x₂, y₂) = (6, 7)
Step 1: Find differences
x₂ - x₁ = 6 - 2 = 4
y₂ - y₁ = 7 - 3 = 4
Step 2: Apply formula
d = √[(4)² + (4)²]
d = √[16 + 16]
d = √32
d ≈ 5.66 units
Answer: √32 or approximately 5.66 units
Example 2: Find distance between (-3, 2) and (5, -4)
Given: (x₁, y₁) = (-3, 2) and (x₂, y₂) = (5, -4)
Calculate differences:
x₂ - x₁ = 5 - (-3) = 5 + 3 = 8
y₂ - y₁ = -4 - 2 = -6
Apply formula:
d = √[(8)² + (-6)²]
d = √[64 + 36]
d = √100
d = 10 units
Answer: 10 units
Special Cases
| Case | Formula | Example |
|---|---|---|
| Horizontal Line (same y-coordinates) | d = |x₂ - x₁| | (2, 5) to (7, 5) d = |7-2| = 5 |
| Vertical Line (same x-coordinates) | d = |y₂ - y₁| | (3, 2) to (3, 8) d = |8-2| = 6 |
Quick Reference: Coordinate Plane
| Concept | Formula/Rule |
|---|---|
| Ordered Pair | (x, y) where x is horizontal, y is vertical |
| Origin | (0, 0) |
| Quadrant I | (+, +) Top Right |
| Quadrant II | (−, +) Top Left |
| Quadrant III | (−, −) Bottom Left |
| Quadrant IV | (+, −) Bottom Right |
| Distance Formula | d = √[(x₂-x₁)² + (y₂-y₁)²] |
💡 Important Tips to Remember
✓ Order matters: (x, y) is NOT the same as (y, x)
✓ X comes first: Remember "x before y" alphabetically
✓ Origin point: Always (0, 0) - the starting point
✓ Quadrants: Count counter-clockwise starting from top-right
✓ Points on axes: Not in any quadrant (one coordinate is 0)
✓ Right/Up: Positive direction; Left/Down: Negative direction
✓ Distance formula: Always gives a positive result
✓ Squaring negatives: Results are always positive in distance formula
✓ Horizontal distance: Subtract x-coordinates only
✓ Vertical distance: Subtract y-coordinates only
🧠 Memory Tricks & Strategies
Ordered Pairs:
"X marks the spot before Y asks why!"
Quadrants:
"All Students Take Calculus" - Quadrants I, II, III, IV
• All positive (Quadrant I)
• Sine positive = y positive (Quadrant II)
• Tangent positive = both negative (Quadrant III)
• Cosine positive = x positive (Quadrant IV)
Simpler Quadrant Rule:
"One is fun, both are plus! Two turns blue, x is minus! Three can't see, both are negative! Four is more, y hits the floor!"
Distance Formula:
"Change in x, change in y, square them both then unify! Add them up and take the root - distance found, compute!"
Plotting Points:
"Run before you rise - x before y, that's wise!"
Movement:
"Right is bright (positive), Left has left (negative) - Up is upper (positive), Down is downer (negative)!"
Master the Coordinate Plane! 📍 📊
Remember: X before Y, and the distance formula is your friend!