🗺️ Linear Functions: Coordinates and Slope
The Treasure Map of Math
Imagine you have a magical treasure map. But this map is special—it uses numbers to tell you EXACTLY where the treasure is hidden!
That’s what the coordinate plane does. It’s like a giant game board that helps us find any spot using just two numbers.
📍 The Coordinate Plane: Your Math Game Board
Think of the coordinate plane as a big piece of graph paper with two roads crossing in the middle.
The Two Roads
- X-axis → The road going left and right (horizontal)
- Y-axis → The road going up and down (vertical)
Where these two roads meet is called the origin. It’s home base! We call it (0, 0).
graph TD A["Origin 0,0"] --> B["Go Right = Positive X"] A --> C["Go Left = Negative X"] A --> D["Go Up = Positive Y"] A --> E["Go Down = Negative Y"]
The Four Neighborhoods
The crossing roads create four areas called quadrants:
| Quadrant | Location | Signs |
|---|---|---|
| I | Upper Right | (+, +) |
| II | Upper Left | (−, +) |
| III | Lower Left | (−, −) |
| IV | Lower Right | (+, −) |
Simple Example: If someone says “go to (3, 2)”, you:
- Start at (0, 0)
- Walk 3 steps RIGHT
- Walk 2 steps UP
- X marks the spot! 🎯
🎯 Plotting Points: Finding Your Spot
Every point on our map has an address with two numbers: (x, y)
The Rule: Always go horizontal first, then vertical.
Think of it like this: You walk on the floor first (x), then take the elevator (y).
Let’s Plot Some Points!
Example 1: Plot (4, 3)
- Start at origin (0, 0)
- Go RIGHT 4 spaces
- Go UP 3 spaces
- Put your dot! ✓
Example 2: Plot (−2, 5)
- Start at origin
- Go LEFT 2 spaces (negative means left!)
- Go UP 5 spaces
- Mark it! ✓
Example 3: Plot (3, −4)
- Start at origin
- Go RIGHT 3 spaces
- Go DOWN 4 spaces (negative means down!)
- Done! ✓
Memory Trick 🧠
“X comes before Y in the alphabet” So x (left/right) comes before y (up/down)!
📏 Distance Formula: How Far Apart?
What if you want to know how far two points are from each other?
Imagine you’re a bird flying from one tree to another in a straight line. The distance formula tells you exactly how far you’d fly!
The Secret: It’s a Triangle!
When you connect two points, you can draw an invisible right triangle. The distance between the points is the long side (called the hypotenuse).
The Formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Don’t let it scare you! Here’s what it means:
- Subtract the x’s → How far apart horizontally?
- Subtract the y’s → How far apart vertically?
- Square both numbers → Multiply each by itself
- Add them together
- Take the square root → Your answer!
Real Example
Find the distance between (1, 2) and (4, 6)
Step 1: Subtract x’s → 4 − 1 = 3 Step 2: Subtract y’s → 6 − 2 = 4 Step 3: Square both → 3² = 9 and 4² = 16 Step 4: Add → 9 + 16 = 25 Step 5: Square root → √25 = 5
The distance is 5 units! 🎉
Why Does This Work?
It’s the Pythagorean theorem in disguise!
- The horizontal distance is one leg (3)
- The vertical distance is the other leg (4)
- The actual distance is the hypotenuse (5)
This is the famous 3-4-5 right triangle!
🎯 Midpoint Formula: Finding the Middle
What if you want to meet a friend EXACTLY halfway between your houses?
The midpoint is the point that’s perfectly in the middle of two other points.
The Formula
$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Translation: Average the x’s, average the y’s!
Real Example
Find the midpoint between (2, 4) and (8, 10)
Step 1: Average the x’s → (2 + 8) ÷ 2 = 10 ÷ 2 = 5 Step 2: Average the y’s → (4 + 10) ÷ 2 = 14 ÷ 2 = 7
The midpoint is (5, 7)! 🎯
Visual Check
Is (5, 7) really in the middle?
- From (2, 4) to (5, 7): go right 3, up 3
- From (5, 7) to (8, 10): go right 3, up 3
Yes! Same distance both ways. ✓
⛰️ What is Slope? The Steepness Story
Have you ever walked up a hill? Some hills are gentle and easy. Others are steep and make your legs tired!
Slope is just a number that tells us how steep a line is.
Slope in Real Life
- Gentle ramp → Small slope (like 0.2)
- Steep staircase → Big slope (like 2)
- Flat floor → Zero slope (0)
- Vertical wall → Undefined slope (can’t walk up that!)
The Four Types of Slope
graph TD A["Types of Slope"] --> B["Positive: Goes UP ↗"] A --> C["Negative: Goes DOWN ↘"] A --> D["Zero: Flat →"] A --> E["Undefined: Vertical ↑"]
| Type | Direction | Example |
|---|---|---|
| Positive | Uphill (left to right) | Walking up a ramp |
| Negative | Downhill (left to right) | Sliding down a slide |
| Zero | Perfectly flat | Walking on a floor |
| Undefined | Straight up/down | Climbing a ladder |
The Meaning of Slope
Slope = Rise ÷ Run
- Rise = How much you go UP (or down)
- Run = How much you go ACROSS
Think of it as: “For every step forward, how many steps up?”
Example: If slope = 2, that means:
- For every 1 step right, you go 2 steps up
- It’s a pretty steep hill!
If slope = ½, that means:
- For every 2 steps right, you go 1 step up
- That’s a gentle slope!
🔢 Calculating Slope: The Math
The Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Where m stands for slope (mathematicians use “m” for slope—nobody knows exactly why!).
Translation
- Top: How much did Y change? (rise)
- Bottom: How much did X change? (run)
Step-by-Step Example
Find the slope between (1, 2) and (5, 10)
Step 1: Label your points
- Point 1: (1, 2) → x₁ = 1, y₁ = 2
- Point 2: (5, 10) → x₂ = 5, y₂ = 10
Step 2: Find the rise (y change)
- y₂ − y₁ = 10 − 2 = 8
Step 3: Find the run (x change)
- x₂ − x₁ = 5 − 1 = 4
Step 4: Divide rise by run
- m = 8 ÷ 4 = 2
The slope is 2!
This means: For every 1 unit you move right, you go up 2 units.
More Examples
Example 2: Points (0, 5) and (3, 5)
- Rise: 5 − 5 = 0
- Run: 3 − 0 = 3
- Slope: 0 ÷ 3 = 0
A horizontal line! Makes sense—no going up or down.
Example 3: Points (2, 1) and (6, -3)
- Rise: −3 − 1 = −4
- Run: 6 − 2 = 4
- Slope: −4 ÷ 4 = −1
Negative slope! The line goes downhill. 🎢
Example 4: Points (4, 2) and (4, 8)
- Rise: 8 − 2 = 6
- Run: 4 − 4 = 0
- Slope: 6 ÷ 0 = Undefined!
You can’t divide by zero! This is a vertical line.
🎓 Quick Summary
| Concept | What It Does | Formula |
|---|---|---|
| Coordinate Plane | The graph paper with x and y axes | — |
| Plotting Points | Finding (x, y) locations | Go right/left, then up/down |
| Distance | How far between two points | √[(x₂−x₁)² + (y₂−y₁)²] |
| Midpoint | Halfway between two points | ((x₁+x₂)/2, (y₁+y₂)/2) |
| Slope | How steep a line is | (y₂−y₁)/(x₂−x₁) |
🚀 You Did It!
You now understand:
- ✅ How to navigate the coordinate plane like a map
- ✅ How to plot any point using (x, y)
- ✅ How to find the distance between two points
- ✅ How to find the exact middle between two points
- ✅ What slope means in real life
- ✅ How to calculate slope from any two points
The coordinate plane is your new superpower. Every graph, every map, every video game uses these exact ideas!
Next time you see a line on a graph, you’ll know its secrets. You can find where it goes, how steep it is, and any point along the way.
That’s the power of coordinates and slope! 📈