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, (y2 - y1)/(x2 - x1), is the same for any two points (x1, y1) and (x2, y2) 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, (x1, y1) and (x2, y2), the slope is the ratio of the change in y to the change in x.
In the equation above, y = x + 1, the slope is the coefficient on x, which is 1.
In the table above, choose any two points to find the slope. The calculation below shows finding the slope using the points (0, 1) and (1, 2).
In the graph above, choose any two points to find the slope. The calculation below shows finding the slope using the points (1, 2) and (2, 3).
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.
8th Grade Math - Linear Relationships Lesson
