Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical
line in a coordinate plane.

Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain
why it will be the same for any two distinct points on the line.

Graph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the
initial value.

Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include
those two sets of points may have different y-intercepts.

Arkansas Academic Standards:
8.EE.B.6

Using a non-vertical or non-horizontal line, show why the slope m is the same between any two distinct points by creating similar triangles

Write the equation y=mx + b for a line through the origin

Be able to write the equation y = mx + b for a line intercepting the vertical axis at b

Arizona Academic Standards:
8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at (0, b).

Common Core State Standards:
Math.8.EE.6 or 8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Georgia Standards of Excellence (GSE):
8.PAR.4.1

Use the equation y = mx (proportional) for a line through the origin to derive the equation y = mx + b (non-proportional) for a line intersecting the vertical axis at b.

Alabama Course of Study Standards:
16

Construct a function to model a linear relationship between two variables.

Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship or
from two points in a table or graph.

Arkansas Academic Standards:
8.F.B.4

Construct a function to model a linear relationship between two quantities:

Determine the rate of change and initial value of the function from:

a verbal description of a relationship

two (x, y) values

a table

a graph

Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values

Arizona Academic Standards:
8.F.B.4

Given a description of a situation, generate a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or a graph. Track how the values of the two quantities change together. Interpret the rate of change and initial value of a linear function in terms of the situation it models, its graph, or its table of values.

Common Core State Standards:
Math.8.F.4 or 8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Georgia Standards of Excellence (GSE):
8.FGR.5.7

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph.

North Carolina - Standard Course of Study:
8.F.4

Analyze functions that model linear relationships.

Understand that a linear relationship can be generalized by y = mx + b.

Write an equation in slope-intercept form to model a linear relationship by determining the rate of change and the initial value, given at least two (x, y) values or a graph.

Construct a graph of a linear relationship given an equation in slope-intercept form.

Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of the slope and y-intercept of its graph or a table of values.

Pennsylvania Core Standards:
M08.B-E.2.1.2

Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane

Pennsylvania Core Standards:
M08.B-E.2.1.3

Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Pennsylvania Core Standards:
CC.2.2.8.C.2

Use concepts of functions to model relationships between quantities.

Pennsylvania Core Standards:
M08.B-F.2.1.1

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.

Georgia Standards of Excellence (GSE):
7.PAR.4.7

Use similar triangles to explain why
the slope, m, is the same between
any two distinct points on a nonvertical line in the coordinate
plane.

8th Grade Math - Linear Relationships Lesson

A linear function is a function which can be written in the form y = mx + b, where m is the slope and b is where the line crosses the y-axis.

The point where a line crosses the y-axis is called the y-intercept. At this point, the x-value is 0.

The table, graph, and equation each represent the same line. The y-intercept of the line is (0, 1).

x

y

-2

-1

-1

0

0

1

1

2

y = x + 1

The slope of a line is also called the rate of change. Given two points on a line, (x_{1}, y_{1}) and (x_{2}, y_{2}), the slope is the ratio of the change in y to the change in x.

A linear function can be represented by an equation, table, or graph.

A linear function can be represented by an equation in the form y = mx + b, where m is the slope and b is the y-intercept.

Values can be substituted into the equation for x to find the corresponding y-values of points on the line. These values can be written in an x,y table or as coordinate pairs.

Example:

First, substitute values for x into the equation and simplify.

x = -2

y = 3(-2) - 1 = -7

x = -1

y = 3(-1) - 1 = -4

x = 0

y = 3(0) - 1 = -1

x = 1

y = 3(1) - 1 = 2

x = 2

y = 3(2) - 1 = 5

x = 3

y = 3(3) - 1 = 8

Then, make an x,y table using the x- and y-values.

x

y

-2

-7

-1

-4

0

-1

1

2

2

5

3

8

Example:

First, plot the points (-2, -7), (-1, -4), (0, -1), (1, 2), (2, 5), and (3, 8) on a coordinate plane.

Then, connect the points to form a line.

Some situations can be represented by a linear equation. Linear equations have a constant rate of change.

In a linear equation of the form y = mx + b, m is the rate of change or slope, and b is the initial value or y-intercept.

Example:

First, find the rate of change. Parking costs $1.50 for each additional hour, so the rate of change is $1.50 per hour.

Next, find the initial value. The airport charges $2 for the first hour of parking, so the initial value is $2.

Write an equation where m = 1.5 and b = 2.

y = 1.5x + 2

Example:

In this situation, the slope, or rate of change, is 2.75, and the y-intercept, or initial value, is 10.

The rate of change is multiplied by the distance, x, to find the cost. So, the slope represents the cost per mile, which is $2.75 per mile.

The initial value is the cost before the letter is carried any distance. So, the y-intercept represents the service fee, which is $10.