3: Linear and Quadratic Functions
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.
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- 3.1: 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.
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- 3.2: 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: \(x=b\).
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- 3.3: 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.
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- 3.4: Quadratic Functions
- In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
Contributors and Attributions
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Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .