Learn on PengiCalifornia Reveal Math, Algebra 1Unit 3: Linear and Nonlinear Functions

3-3 Slope-Intercept Form

In this Grade 9 lesson from California Reveal Math, Algebra 1 (Unit 3), students learn to write, rewrite, and graph linear equations in slope-intercept form y = mx + b, where m represents slope and b represents the y-intercept. The lesson covers converting standard form equations like -22x + 8y = 4 into slope-intercept form using properties of equality, as well as modeling real-world situations with linear equations. Students also explore how the parameters m and b affect the graph of a linear function.

Section 1

Notation for Slope

Property

The slope of a line is given by

m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}, \quad \Delta x \neq 0

The symbol $\Delta$ (delta) is used in mathematics to denote change in.

Examples

If the change in y, $\Delta y$ , is 6 and the change in x, $\Delta x$ , is 2, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3$ .
A line moves 4 units down ( $Δy = -4$ ) for every 10 units it moves to the right ( $Δx = 10$ ). The slope is $m = \frac{-4}{10} = -\frac{2}{5}$ .
For a horizontal line, the y-coordinate never changes, so $\Delta y = 0$ . This means the slope $m = \frac{0}{\Delta x} = 0$ , no matter the change in x.

Explanation

This is the official shorthand for slope. The letter $m$ stands for slope, and the Greek letter delta ( $\Delta$ ) is a compact way to write 'change in'. So, $m = \frac{\Delta y}{\Delta x}$ is just a neat way of writing slope equals change in y over change in x.

Section 2

Linear Data Has Constant First Differences

Property

When data follows a linear pattern, the first differences between consecutive $y$ -values are constant.

For data points with evenly spaced $x$ -values, if $\Delta y = y_{i+1} - y_i$ is the same for all consecutive pairs, then the data can be modeled by a linear function.

Section 3

Slope-intercept form

Property

A linear equation written in the form

y = mx + b

is said to be in slope-intercept form. The coefficient $m$ is the slope of the graph, and $b$ is the $y$ -intercept.

Examples

The equation $y = 3x + 5$ is in slope-intercept form. The slope is $3$ and the $y$ -intercept is $(0, 5)$ .
For $y = -2x - 1$ , the slope is $-2$ and the $y$ -intercept is $(0, -1)$ .
In the equation $y = \frac{1}{2}x + 4$ , the slope is $\frac{1}{2}$ and the $y$ -intercept is $(0, 4)$ .

Explanation

This form is a recipe for drawing a line. The ' $b$ ' tells you your starting point on the y-axis, and the ' $m$ ' (slope) gives you directions on how steep to draw the line from there.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Notation for Slope

Property

The slope of a line is given by

m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}, \quad \Delta x \neq 0

The symbol $\Delta$ (delta) is used in mathematics to denote change in.

Examples

If the change in y, $\Delta y$ , is 6 and the change in x, $\Delta x$ , is 2, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3$ .
A line moves 4 units down ( $Δy = -4$ ) for every 10 units it moves to the right ( $Δx = 10$ ). The slope is $m = \frac{-4}{10} = -\frac{2}{5}$ .
For a horizontal line, the y-coordinate never changes, so $\Delta y = 0$ . This means the slope $m = \frac{0}{\Delta x} = 0$ , no matter the change in x.

Explanation

Section 2

Linear Data Has Constant First Differences

Property

When data follows a linear pattern, the first differences between consecutive $y$ -values are constant.

For data points with evenly spaced $x$ -values, if $\Delta y = y_{i+1} - y_i$ is the same for all consecutive pairs, then the data can be modeled by a linear function.

Section 3

Slope-intercept form

Property

A linear equation written in the form

y = mx + b

is said to be in slope-intercept form. The coefficient $m$ is the slope of the graph, and $b$ is the $y$ -intercept.

Examples

The equation $y = 3x + 5$ is in slope-intercept form. The slope is $3$ and the $y$ -intercept is $(0, 5)$ .
For $y = -2x - 1$ , the slope is $-2$ and the $y$ -intercept is $(0, -1)$ .
In the equation $y = \frac{1}{2}x + 4$ , the slope is $\frac{1}{2}$ and the $y$ -intercept is $(0, 4)$ .

Explanation

This form is a recipe for drawing a line. The ' $b$ ' tells you your starting point on the y-axis, and the ' $m$ ' (slope) gives you directions on how steep to draw the line from there.

Overview

Lesson overview

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

Overview

Section 1

Notation for Slope

Property

The slope of a line is given by

m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}, \quad \Delta x \neq 0

The symbol $\Delta$ (delta) is used in mathematics to denote change in.

Examples

If the change in y, $\Delta y$ , is 6 and the change in x, $\Delta x$ , is 2, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3$ .
A line moves 4 units down ( $Δy = -4$ ) for every 10 units it moves to the right ( $Δx = 10$ ). The slope is $m = \frac{-4}{10} = -\frac{2}{5}$ .
For a horizontal line, the y-coordinate never changes, so $\Delta y = 0$ . This means the slope $m = \frac{0}{\Delta x} = 0$ , no matter the change in x.

Explanation

Section 2

Linear Data Has Constant First Differences

Property

When data follows a linear pattern, the first differences between consecutive $y$ -values are constant.

For data points with evenly spaced $x$ -values, if $\Delta y = y_{i+1} - y_i$ is the same for all consecutive pairs, then the data can be modeled by a linear function.

Section 3

Slope-intercept form

Property

A linear equation written in the form

y = mx + b

is said to be in slope-intercept form. The coefficient $m$ is the slope of the graph, and $b$ is the $y$ -intercept.

Examples

The equation $y = 3x + 5$ is in slope-intercept form. The slope is $3$ and the $y$ -intercept is $(0, 5)$ .
For $y = -2x - 1$ , the slope is $-2$ and the $y$ -intercept is $(0, -1)$ .
In the equation $y = \frac{1}{2}x + 4$ , the slope is $\frac{1}{2}$ and the $y$ -intercept is $(0, 4)$ .

Explanation

This form is a recipe for drawing a line. The ' $b$ ' tells you your starting point on the y-axis, and the ' $m$ ' (slope) gives you directions on how steep to draw the line from there.