📈 Graphing Proportional Relationships
Drawing the connection
🎯 Introduction
Graphs are like pictures of relationships! When you graph a proportional relationship, something magical happens - you always get a perfectly straight line through the origin!
Proportional relationships create straight lines that pass through the point (0, 0) - called the origin.
📊 Visual Guide
💡 Key Concepts
Graphs of Proportional Relationships
When you plot points from a proportional relationship and connect them, you get a straight line. This line ALWAYS goes through (0, 0).
📌 Real Example:
If you graph carpet cost vs. square feet, and 0 sq ft costs 15, 20 sq ft costs $30... these points form a straight line through the origin.
💭 Imagine drawing a line from the corner of the graph (0,0) straight up and to the right. That's what a proportional relationship looks like!
Why (0, 0) Matters
The origin (0, 0) MUST be on the line because: if you have 0 of one thing, you have 0 of the other. Zero times anything is zero!
📌 Real Example:
0 square feet of carpet costs 0 pay. 0 candies cost $0. Makes sense, right?
💭 If a line doesn't go through (0, 0), the relationship is NOT proportional.
Finding k from a Graph
You can find k by picking ANY point on the line and dividing: k = y ÷ x. You can also look at the 'steepness' (slope) of the line.
📌 Real Example:
If the point (10, 15) is on the line, then k = 15 ÷ 10 = 1.5
💭 Steeper lines have bigger k values. Flatter lines have smaller k values.
Plotting Points
To graph a proportional relationship: 1) Make a table of values, 2) Plot each point (x, y), 3) Connect with a straight line through (0, 0).
📌 Real Example:
For the grape/peach juice recipe (5 grape → 2 peach): Plot (0,0), (5,2), (10,4), (15,6). Connect the dots!
💭 Think of it like connect-the-dots, but the dots are always in a straight line.
📝 Step-by-Step Examples
Example 1: Graphing from a table
📋 Problem: Graph the carpet cost relationship: 10 sq ft = 30, 40 sq ft = $60
Set up your axes
x-axis: square feet (0 to 50), y-axis: cost in dollars (0 to 70)
Always start with (0, 0)
Plot the origin: 0 sq ft costs $0
Plot each point from the table
Plot (10, 15), (20, 30), (40, 60)
Connect with a straight line
Draw a line through all points, extending through (0,0)
Label your graph
Title: 'Carpet Cost vs. Square Feet', label each axis
✅ Answer: A straight line through (0,0), (10,15), (20,30), (40,60)
Example 2: Finding k from a graph
📋 Problem: A graph shows points (5, 2), (10, 4), and (15, 6). Find k.
Pick any point (except the origin)
Let's use (10, 4)
Divide y by x
k = 4 ÷ 10 = 0.4
Check with another point
(5, 2): k = 2 ÷ 5 = 0.4 ✓
✅ Answer: k = 0.4 or 2/5
Example 3: Is this graph proportional?
📋 Problem: A graph shows a straight line, but it passes through (0, 5) instead of (0, 0). Is it proportional?
Check the key feature
Does the line pass through (0, 0)?
Analyze
No! It passes through (0, 5)
Conclude
NOT proportional - even though it's straight, it doesn't go through the origin
✅ Answer: No, it's NOT proportional because it doesn't pass through (0, 0)
⚠️ Common Mistakes to Avoid
❌ Forgetting to include (0, 0)
Why it's wrong: Even if (0, 0) isn't in your table, it MUST be on the line for proportional relationships.
✅ How to avoid: Always start by plotting (0, 0), then add your other points.
❌ Drawing a curved line
Why it's wrong: Proportional relationships are ALWAYS straight lines. If your points don't line up straight, check your math!
✅ How to avoid: Use a ruler! If points don't line up, you made a calculation error somewhere.
❌ Mixing up x and y coordinates
Why it's wrong: (10, 15) means x=10, y=15. Swapping them gives the wrong point!
✅ How to avoid: Remember: (x, y) means (horizontal, vertical). Go right first, then up.
Graphs are super powerful! 📊 Now you can SEE relationships, not just calculate them. Plus, you can use a graph to estimate values that aren't in your table!