Linear Functions

Student Summary

Suppose a car is traveling at 30 miles per hour. The relationship between the time in hours and the distance in miles is a proportional relationship.

We can represent this relationship with an equation of the form d=30td = 30t, where distance is a function of time (since each input of time has exactly one output of distance).

Or we could write the equation t=130dt = \frac{1}{30} d instead, where time is a function of distance (since each input of distance has exactly one output of time).

More generally, if we represent a linear function with an equation like y=mx+by = mx + b, then bb is the initial value (which is 0 for proportional relationships), and mm is the rate of change of the function.

If mm is positive, the function is increasing.

If mm is negative, the function is decreasing.

If we represent a linear function in a different way, say with a graph, we can use what we know about graphs of lines to find the mm and bb values and, if needed, write an equation.

Visual / Anchor Chart

Standards

Building On
8.EE.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

8.EE.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.

Addressing
8.F.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.

8.F.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Recognize examples of functions that are linear and non-linear.

8.F.2

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).