Texas Essential Knowledge and Skills (TEKS):
8.4.A

use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (y_{2} - y_{1})/(x_{2} - x_{1}), is the same for any two points (x_{1}, y_{1}) and (x_{2}, y_{2}) on the same line;

Texas Essential Knowledge and Skills (TEKS):
8.4.C

use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

Texas Essential Knowledge and Skills (TEKS):
8.5.B

represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0;

Texas Essential Knowledge and Skills (TEKS):
8.5.I

write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

Florida - Benchmarks for Excellent Student Thinking:
MA.8.AR.3.2

Given a table, graph or written description of a linear relationship, determine the slope.

Florida - Benchmarks for Excellent Student Thinking:
MA.8.AR.3.3

Given a table, graph or written description of a linear relationship, write an equation in slope-intercept form.

Florida - Benchmarks for Excellent Student Thinking:
MA.8.AR.3.4

Given a mathematical or real-world context, graph a two-variable linear equation from a written description, a table or an equation in slope-intercept form.

Florida - Benchmarks for Excellent Student Thinking:
MA.8.AR.3.5

Given a real-world context, determine and interpret the slope and y-intercept of a two-variable linear equation from a written description, a table, a graph or an equation in slope-intercept form.

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.