Recall that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data.
- Section 2.1: Linear Functions
- The ordered pairs given by a linear function represent points on a line. Linear functions can be represented in words, function notation, tabular form, and graphical form. The rate of change of a linear function is also known as the slope. An equation in the slope-intercept form of a line includes the slope and the initial value of the function. The initial value, or y-intercept, is the output value when the input of a linear function is zero.
- Section 2.2: Graphs of Linear Functions
- Linear functions may be graphed by plotting points or by using the y-intercept and slope. Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. The y-intercept and slope of a line may be used to write the equation of a line. The x-intercept is the point at which the graph of a linear function crosses the x-axis. Horizontal lines are written like: \(f(x)=b\). Vertical lines are written like: \(x=b\).
- Section 2.3: Modeling with Linear Functions
- We can use the same problem strategies that we would use for any type of function. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. Draw a diagram, where appropriate. Check for reasonableness of the answer. Linear models may be built by identifying or calculating the slope and using the y-intercept. The x-intercept may be found by setting y=0, which is setting the expression mx+b equal to 0.