8.PAR.4.1 8.FGR.5.5 8.FGR.5.6 8.FGR.5.7 8.FGR.5.8

Linear Relationships 8th Grade

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.
Georgia Standards of Excellence (GSE): 8.FGR.5.5
Write and explain the equations y = mx + b (slope-intercept form), Ax + By = C (standard form), and (y - y₁) = m(x-x₁) (point-slope form) as defining a linear function whose graph is a straight line to reveal and explain different properties of the function.
Georgia Standards of Excellence (GSE): 8.FGR.5.6
Write a linear function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
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.
Georgia Standards of Excellence (GSE): 8.FGR.5.8
Explain the meaning of 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.

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.

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

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).

\begin{array}{rcl} m & = & \frac{2 - 1}{1 - 0} \\ = & \frac{1}{1} \\ = & 1 \end{array}

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).

\begin{array}{rcl} m & = & \frac{3 - 2}{2 - 1} \\ = & \frac{1}{1} \\ = & 1 \end{array}

Linear Relationships

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.

Georgia Standards of Excellence (GSE): 8.FGR.5.5

Write and explain the equations y = mx + b (slope-intercept form), Ax + By = C (standard form), and (y - y₁) = m(x-x₁) (point-slope form) as defining a linear function whose graph is a straight line to reveal and explain different properties of the function.

Georgia Standards of Excellence (GSE): 8.FGR.5.6

Write a linear function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

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.

Georgia Standards of Excellence (GSE): 8.FGR.5.8

Explain the meaning of 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.

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