Graph Proportional
Relationships
Learn to graph proportional relationships, interpret the unit rate as slope, and compare proportional relationships in different forms.
What Is a Proportional Relationship?
A proportional relationship is one where two quantities always have the same ratio. If you double one quantity, the other doubles too. The general equation is:
The constant \( k \) tells you how much \( y \) changes for every 1 unit increase in \( x \). This is also called the unit rate.
Seeing It on a Graph
Every proportional relationship graphs as a straight line through the origin \((0,0)\). The steeper the line, the greater the unit rate.
Unit Rate = Slope
In a proportional relationship \( y = kx \), the constant \( k \) is both the unit rate and the slope of the line.
Why This Matters
Proportional relationships are everywhere. Anytime you see a constant rate, you are looking at a proportional relationship.
The Core Rule
The right graph starts at \((0, 1)\), not \((0, 0)\). That means it has a y-intercept, so it is not proportional. Its equation is \( y = mx + b \) with \( b \neq 0 \).
Reading Slope from a Graph
Proportional vs. Linear
| Proportional | Linear (Not Proportional) | |
|---|---|---|
| Equation | \( y = kx \) | \( y = mx + b \), \( b \neq 0 \) |
| Graph | Line through (0,0) | Line through (0,b) |
| y-intercept | 0 | \( b \neq 0 \) |
| Ratio \( y/x \) | Always constant | Changes with \( x \) |
| Example | \( y = 3x \) | \( y = 3x + 2 \) |
Comparing in Different Forms
Proportional relationships can be given as a graph, a table, an equation, or a verbal description. To compare them, find the unit rate \( k \) from each form.
How to Apply It
Example 1: Graphing from a Table
A runner logs the following data. Graph the relationship and find the unit rate.
| Time (min) | Distance (mi) |
|---|---|
| 0 | 0 |
| 5 | 0.5 |
| 10 | 1.0 |
| 20 | 2.0 |
Example 1: The Graph
Example 2: Comparing Two Rates
Two cars are traveling at constant speeds. Which car is faster?
where \( d \) is in miles and \( t \) is in hours.
| Hours | Miles |
| 2 | 120 |
| 4 | 240 |
Example 3: Graph vs. Equation
Store A charges by the equation \( C = 4.50n \). Store B's pricing is shown on the graph. Which store has the lower unit price?
Common Mistakes
Summary Table
| Concept | Key Idea |
|---|---|
| Proportional Relationship | \( y = kx \), passes through \((0,0)\) |
| Unit Rate / Slope | \( k = \dfrac{y}{x} = \dfrac{\text{rise}}{\text{run}} \) |
| From a Table | Divide any \( y \) by its \( x \) |
| From a Graph | Pick a point, compute \( k = y/x \) |
| From an Equation | Coefficient of \( x \) is \( k \) |
| Comparing | Larger \( k \) = steeper line = greater rate |
Question 1 of 5
A proportional relationship is graphed as a straight line. Which point must the line pass through?
Question 2 of 5
The equation \( y = 7x \) represents a proportional relationship. What is the unit rate?
Question 3 of 5
A graph of a proportional relationship passes through the point \((4, 12)\). What is the slope of the line?
Question 4 of 5
Runner A runs at \( d = 8t \) (miles in hours). Runner B runs 42 miles in 6 hours at a constant speed. Who is faster?
Question 5 of 5
Which equation does NOT represent a proportional relationship?