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Objectives
Objectives
Provide a definition of linear functions.
Identify linear functions from graphs, tables, and equations
Definition
Definition
A linear function is a function whose graph is a straight path. Mathematically, we say a graph whose points represents a constant rate of change. The phrase “constant rate of change” is synonymous with slope and means that the rate of change will be the same between any two points of the graph. The slope formula can be expressed in a variety of ways.
Let the variable m represent the slope of a line, and let there be two points whose coordinates are and , then we can define the slope formula as:
Each of the formulas are the same just expressed differently. It is important to be familiar with all the ways that slope is expressed so that we can better solve real world word problems that use phrases like “constant rate of change” or “rise over run”.
Linear Graphs
Linear Graphs
The following images are examples of linear graphs:
The following images are graphs of non–linear functions.
In summary, we can identify linear functions by their graphs. The is either a straight line or a straight path of points. If there is any change in direction or “curving” of the path, then the graph is non – linear. Now that we know how to identify linear functions by graphs, we will now turn our attention towards identifying linear functions by data sets and by equations.
Linear Data Sets
Linear Data Sets
Linear data sets can be represented as a list of points or as a table of values. Either way. We can identify whether a data set is linear or not by evaluating the slope between every pair of points. If the slope is always the same, then it is a linear data set. If the slope is different in just one instance, then the set is non – linear.
Analyze the data by evaluating the slope between each pair of points. Calculate the difference in y – values over the difference in x – values. If the table always generates the same slope, then it represents a linear function.
Linear Equations
Linear Equations
Another way to identify a linear function is by its equation. Linear equations can be expressed in a variety of formats. However, the simplest way to contextualize linear equations is as a first–degree equation. A first–degree equation is one where the variables in the equation have an exponent of positive one and none of the variables multiply one another.
Examples:
Non–linear equations have exponents not equal to positive one or can be equations containing the products of different variables.
Examples:
In summary, we can identify linear functions by their equations if the variables have exponents of positive one, the variables are not multiplied together, and the variables do not appear in denominators, in exponents, or in radical signs.
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