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+bf(x) = mx + b, where mm is the constant rate of change known as the slope and bb is the yy-intercept of the function. A linear function can be graphed using the slope and yy-intercept and by considering transformations of the parent function f(x)=xf(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+bf(x) = mx + b, where mm is the slope and bb is the yy-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.