Learn on PengienVision, Mathematics, Grade 8Chapter 2: Analyze and Solve Linear Equations

Lesson 8: Understand the y-Intercept of a Line

In this Grade 8 enVision Mathematics lesson from Chapter 2, students learn how to identify the y-intercept of a linear graph and explain what it represents in real-world contexts. Using examples like bowling costs and robotic assembly lines, students practice finding where a line crosses the y-axis and interpreting whether the y-intercept is positive, negative, or zero. Students also discover that proportional relationships always have a y-intercept of 0, passing through the origin.

Section 1

Intercepts of a Line

Property

Each of the points at which a line crosses the $x$ -axis and the $y$ -axis is called an intercept of the line.

The $x$ -intercept is the point, $(a, 0)$ , where the graph crosses the $x$ -axis. The $x$ -intercept occurs when $y$ is zero.

The $y$ -intercept is the point, $(0, b)$ , where the graph crosses the $y$ -axis. The $y$ -intercept occurs when $x$ is zero.

Section 2

Real-World Meaning of Slope and Y-Intercept

Property

In a real-world context described by the equation $y = mx + b$ :

The slope ( $m$ ) represents the rate of change. It tells you how much the dependent variable ( $y$ ) changes for every one-unit increase in the independent variable ( $x$ ).
The y-intercept ( $b$ ) represents the initial value or starting point. It is the value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is zero.

Examples

Section 3

Graphing Proportional Variables

Property

When graphed, the relationship between two proportional variables has two key characteristics:

The graph is a straight line.
The graph passes through the origin, which is the point $(0, 0)$ .

These features occur because the rate of change is constant and because if one variable is zero, the other must also be zero.

Examples

A graph shows the cost of bulk almonds. The point $(4, 24)$ is on the line, meaning 4 pounds cost 24 dollars. Since the graph is a line through the origin, the unit price is constant: $\frac{24}{4} = 6$ dollars per pound.
The graph of a monthly bus pass cost is a horizontal line at $y=50$ . This is not proportional to the number of rides because it does not pass through $(0,0)$ and the cost is constant regardless of the number of rides.
A caterer's fee is shown on a graph that is a straight line through $(0,0)$ and $(10, 150)$ . The relationship is proportional. The unit rate (cost per person) is $\frac{150}{10} = 15$ dollars per person. For 30 people, the cost would be $30 \times 15 = 450$ dollars.

Explanation

Think of a proportional graph as a perfectly straight ramp that starts right at the ground. It is straight because the steepness (the rate) never changes, and it starts at $(0,0)$ because zero input means zero output, like working 0 hours earns 0 dollars.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Intercepts of a Line

Property

Each of the points at which a line crosses the $x$ -axis and the $y$ -axis is called an intercept of the line.

The $x$ -intercept is the point, $(a, 0)$ , where the graph crosses the $x$ -axis. The $x$ -intercept occurs when $y$ is zero.

The $y$ -intercept is the point, $(0, b)$ , where the graph crosses the $y$ -axis. The $y$ -intercept occurs when $x$ is zero.

Section 2

Real-World Meaning of Slope and Y-Intercept

Property

In a real-world context described by the equation $y = mx + b$ :

The slope ( $m$ ) represents the rate of change. It tells you how much the dependent variable ( $y$ ) changes for every one-unit increase in the independent variable ( $x$ ).
The y-intercept ( $b$ ) represents the initial value or starting point. It is the value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is zero.

Examples

Section 3

Graphing Proportional Variables

Property

When graphed, the relationship between two proportional variables has two key characteristics:

The graph is a straight line.
The graph passes through the origin, which is the point $(0, 0)$ .

These features occur because the rate of change is constant and because if one variable is zero, the other must also be zero.

Examples

A graph shows the cost of bulk almonds. The point $(4, 24)$ is on the line, meaning 4 pounds cost 24 dollars. Since the graph is a line through the origin, the unit price is constant: $\frac{24}{4} = 6$ dollars per pound.
The graph of a monthly bus pass cost is a horizontal line at $y=50$ . This is not proportional to the number of rides because it does not pass through $(0,0)$ and the cost is constant regardless of the number of rides.
A caterer's fee is shown on a graph that is a straight line through $(0,0)$ and $(10, 150)$ . The relationship is proportional. The unit rate (cost per person) is $\frac{150}{10} = 15$ dollars per person. For 30 people, the cost would be $30 \times 15 = 450$ dollars.

Explanation

Overview

Lesson overview

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

Overview

Section 1

Intercepts of a Line

Property

Each of the points at which a line crosses the $x$ -axis and the $y$ -axis is called an intercept of the line.

The $x$ -intercept is the point, $(a, 0)$ , where the graph crosses the $x$ -axis. The $x$ -intercept occurs when $y$ is zero.

The $y$ -intercept is the point, $(0, b)$ , where the graph crosses the $y$ -axis. The $y$ -intercept occurs when $x$ is zero.

Section 2

Real-World Meaning of Slope and Y-Intercept

Property

In a real-world context described by the equation $y = mx + b$ :

The slope ( $m$ ) represents the rate of change. It tells you how much the dependent variable ( $y$ ) changes for every one-unit increase in the independent variable ( $x$ ).
The y-intercept ( $b$ ) represents the initial value or starting point. It is the value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is zero.

Examples

Section 3

Graphing Proportional Variables

Property

When graphed, the relationship between two proportional variables has two key characteristics:

The graph is a straight line.
The graph passes through the origin, which is the point $(0, 0)$ .

These features occur because the rate of change is constant and because if one variable is zero, the other must also be zero.

Examples

A graph shows the cost of bulk almonds. The point $(4, 24)$ is on the line, meaning 4 pounds cost 24 dollars. Since the graph is a line through the origin, the unit price is constant: $\frac{24}{4} = 6$ dollars per pound.
The graph of a monthly bus pass cost is a horizontal line at $y=50$ . This is not proportional to the number of rides because it does not pass through $(0,0)$ and the cost is constant regardless of the number of rides.
A caterer's fee is shown on a graph that is a straight line through $(0,0)$ and $(10, 150)$ . The relationship is proportional. The unit rate (cost per person) is $\frac{150}{10} = 15$ dollars per person. For 30 people, the cost would be $30 \times 15 = 450$ dollars.