Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 3: Graphing and Writing Linear Equations

Lesson 3: Graphing Proportional Relationships

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, students learn to identify, graph, and write proportional relationships using the direct variation equation y = mx. Students explore how the slope m represents the unit rate and use similar triangles to derive why every proportional relationship passes through the origin. The lesson connects ratio tables, coordinate graphs, and algebraic equations to build a complete understanding of proportional relationships.

Section 1

Identifying Proportional Relationships

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

Using a table: Test for equivalent ratios. For any pair of corresponding quantities $(x, y)$ , the ratio $\frac{y}{x}$ must be the same for all non-zero pairs.
Using a graph: The graph of the relationship must be a straight line that passes through the origin $(0, 0)$ .

Examples

A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through $(0,0)$ and $(1,2)$ , showing it's proportional.
A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at $(0,10)$ , not the origin, so it is not proportional.

Explanation

Think of it like a recipe. If you double the flour, you must double the sugar. A proportional relationship means two quantities scale up or down together at a steady rate. The graph is a straight line starting from zero because zero input means zero output.

Section 2

Understanding Graphs of Proportional Relationships

Property

If quantities $y$ and $x$ are in proportion then the graph of pairs $(x, y)$ in this relation will be a straight line through the origin.
That line is characterized by the assertion that $\frac{y}{x}$ is constant, and in fact, is the constant of proportionality.
In terms of the graph, we call this its slope.

Examples

If you earn 20 dollars per hour, your earnings $E$ are related to hours $h$ by $E=20h$ . The graph is a straight line passing through $(0,0)$ , $(1,20)$ , and $(3,60)$ .
The data points $(2, 8)$ , $(3, 12)$ , and $(5, 20)$ all lie on a straight line through the origin. The constant ratio $\frac{y}{x} = 4$ is the slope of the line.
The weight of an object in kilograms ( $k$ ) is proportional to its weight in pounds ( $p$ ). Since $k \approx 0.45p$ , the graph of this relationship is a line through the origin with a slope of approximately 0.45.

Explanation

A proportional relationship always starts at $(0, 0)$ because zero of one thing means zero of the other. The graph is a straight line, and its steepness, or slope, is simply the unit rate. A steeper line means a higher rate.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Identifying Proportional Relationships

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

Using a table: Test for equivalent ratios. For any pair of corresponding quantities $(x, y)$ , the ratio $\frac{y}{x}$ must be the same for all non-zero pairs.
Using a graph: The graph of the relationship must be a straight line that passes through the origin $(0, 0)$ .

Examples

A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through $(0,0)$ and $(1,2)$ , showing it's proportional.
A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at $(0,10)$ , not the origin, so it is not proportional.

Explanation

Section 2

Understanding Graphs of Proportional Relationships

Property

Examples

If you earn 20 dollars per hour, your earnings $E$ are related to hours $h$ by $E=20h$ . The graph is a straight line passing through $(0,0)$ , $(1,20)$ , and $(3,60)$ .
The data points $(2, 8)$ , $(3, 12)$ , and $(5, 20)$ all lie on a straight line through the origin. The constant ratio $\frac{y}{x} = 4$ is the slope of the line.
The weight of an object in kilograms ( $k$ ) is proportional to its weight in pounds ( $p$ ). Since $k \approx 0.45p$ , the graph of this relationship is a line through the origin with a slope of approximately 0.45.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Identifying Proportional Relationships

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

Using a table: Test for equivalent ratios. For any pair of corresponding quantities $(x, y)$ , the ratio $\frac{y}{x}$ must be the same for all non-zero pairs.
Using a graph: The graph of the relationship must be a straight line that passes through the origin $(0, 0)$ .

Examples

A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through $(0,0)$ and $(1,2)$ , showing it's proportional.
A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at $(0,10)$ , not the origin, so it is not proportional.

Explanation

Section 2

Understanding Graphs of Proportional Relationships

Property

Examples

If you earn 20 dollars per hour, your earnings $E$ are related to hours $h$ by $E=20h$ . The graph is a straight line passing through $(0,0)$ , $(1,20)$ , and $(3,60)$ .
The data points $(2, 8)$ , $(3, 12)$ , and $(5, 20)$ all lie on a straight line through the origin. The constant ratio $\frac{y}{x} = 4$ is the slope of the line.
The weight of an object in kilograms ( $k$ ) is proportional to its weight in pounds ( $p$ ). Since $k \approx 0.45p$ , the graph of this relationship is a line through the origin with a slope of approximately 0.45.