Unit 4: Linear Functions
In this unit, you will explore linear functions in depth. Linear functions are a great place to start our in-depth exploration of functions because they are fairly straightforward to master, and many people have learned about them in past math courses. In our discovery of linear functions, we will work with tables, equations, graphs, and words to describe the behavior and key characteristics of linear functions. You will also learn about an important concept called the rate of change. One of the most important key characteristics of a linear function is its rate of change. If you continue on to study calculus, you will learn about the rate of change in depth. In addition to the rate of change, you will discover that some observable phenomena behave linearly. We can build a model using a linear function to represent their behavior and make predictions.
Completing this unit should take you approximately 2 hours.
Upon successful completion of this unit, you will be able to:
- graph points and lines on the Cartesian coordinate plane;
- represent a linear function using words, tables, graphs, and function notation;
- describe the behavior of a linear function over an interval;
- interpret the slope of a line using words;
- construct the equation of a linear function given two points, a table, or words;
- sketch the graph of a linear function using points, the slope and intercept, equations, and transformations;
- construct the equation for a linear function given a graph;
- make predictions using linear models constructed from words and data;
- interpret whether scatter diagrams represent a linear relation;
- use a graphing utility to construct a linear regression line given a data set;
- analyze how changes in data affect a regression line; and
- distinguish between interpolation and extrapolation.
4.1: Linear Functions
In this introductory section, you will practice representing a linear function using words, function notation, tables, and graphs. We will also introduce the language and techniques used to determine whether the graph of a linear function is increasing, decreasing, or constant.
- Slope is one of the most important characteristics of a linear equation. We can use it to determine whether a linear function is increasing, decreasing, or constant, and we can also use it to graph linear functions quickly and easily. When a linear function represents a real-world application, the slope tells us important information about what is being modeled.
The equation of a linear function is different than that of the linear equations we solved in Unit 1 because it contains two variables. The two variables typically used in a linear function are and . The values in the equation represent the inputs to the function, and the values represent the outputs. In addition to the variables and , the linear function contains a slope and a constant called the y-intercept.
Now, you will build on your knowledge of writing equations of linear functions, drawing their graphs, and performing transformations by exploring parallel and perpendicular lines. You will learn how to identify whether lines are parallel or perpendicular given their equations and graphs. You will also learn how to define the equation of a line parallel or perpendicular to another line given an equation and a point.
4.2: Modeling with Linear Functions
This section introduces a basic problem-solving method that will help you navigate the task of creating a linear model given verbal descriptions.
Now, we will look at a system of two linear equations. There are three different types of solutions for a system of linear equations. We will explore this topic in-depth later in the course.
4.3: Fitting Linear Models to Data
- In this section, you will learn a convenient application for linear functions. Given a set of data points, you will be able to determine whether it is linear, and if it is, you will learn how to determine its equation using an online graphing calculator. It's fun when you can finally see the practical applications in math.