# Lesson 8

Linear Functions

### Lesson Narrative

This is the first of three lessons about linear functions. Students are already familiar with linear equations and their graphs from previous units.

In the first activity, students see that a proportional relationship between two quantities can be viewed as a function. They see that either quantity can be chosen as the independent variable and that the only difference in the equation and the graph is the constant of proportionality, which is visible on the graph as the slope of the line through the origin.

In the next activities, students investigate and make connections between linear functions as represented by graphs, descriptions, and by the equation $$y=mx+b$$. They interpret the slope of the line as the rate of change $$m$$ of the dependent variable with respect to the independent variable and the vertical intercept of the line as the initial value $$b$$ of the function. Students also compare properties of linear functions represented in different ways to determine, for example, which function has the greater rate of change. Consider using the optional activity if students need more practice comparing linear functions represented in different ways.

### Learning Goals

Teacher Facing

• Comprehend that any linear function can be represented by an equation in the form $y=mx+b$, where $m$ and $b$ are rate of change and initial value of the function, respectively.
• Coordinate (orally and in writing) the graph of a linear function and its rate of change and initial value.

### Student Facing

Let’s investigate linear functions.

### Student Facing

• I can determine whether a function is increasing or decreasing based on whether its rate of change is positive or negative.
• I can explain in my own words how the graph of a linear function relates to its rate of change and initial value.