# Linear Functions and Modeling

## Overview

### Description

Linear functions model many real-world situations that involve constant rates of change, such as speed or cost. Linear functions can be written in the form $f(x) = mx + b$, where $m$ is the constant rate of change known as the slope and $b$ is the $y$-intercept of the function. A linear function can be graphed using the slope and $y$-intercept and by considering transformations of the parent function $f(x) = x$.

Scatterplots can be used to understand the relationship between two variables. A process called linear regression is used to find an equation that best approximates a linear relationship between the scatterplot data. This equation, known as the line of best fit, allows for the prediction of data values.

### At A Glance

• A linear function is written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept of the function. The slope is also called the rate of change.
• A linear function has a constant rate of change equal to the slope of its graph.
• A translation of the graph of a linear function is a shift vertically or horizontally.
• A stretch or compression of the graph of a linear function is a pull away from an axis or a push toward an axis. A reflection of the graph of a linear function is a flip across a line.
• Some linear functions are the result of multiple transformations of the linear parent function.
• Data can be represented in two variables using a scatterplot.
• A scatterplot can be used to visualize trends in data that indicate a positive, negative, or no linear relationship between two variables. The type of relationship between the variables is indicated by the slope of the line of best fit.
• Linear regression is the process used to find the equation of a line of best fit that approximates the closest linear relationship between two variables. The correlation coefficient indicates the strength of the linear fit to the data.
• Technology can be used to generate a line of best fit.
• A line of best fit can be used to predict data values.