Understanding Linear Functions: Definition, Graphing, and Applications

Linear Function

A linear function is a mathematical function that represents a straight line on a graph

A linear function is a mathematical function that represents a straight line on a graph. It is defined by the equation y = mx + b, where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.

The slope, m, determines the steepness or the direction of the line. A positive slope means the line goes up from left to right, while a negative slope means the line goes down from left to right. The value of m can be determined by calculating the change in y divided by the change in x between any two points on the line.

The y-intercept, b, represents the point where the line crosses the y-axis. It is the value of y when x is equal to zero. To find the y-intercept, substitute x = 0 into the equation and solve for y.

To graph a linear function, you can plot the y-intercept as one point on the graph, and then use the slope to find additional points. You can do this by using the slope to find the change in y and change in x from the y-intercept, and then plot another point. Once you have two points, you can draw a straight line through them to represent the linear function.

Linear functions are useful in many areas of mathematics, science, and economics. They are used to solve problems involving rates of change, such as calculating speeds, growth rates, or predicting future values based on past data. They are also commonly used in algebra to model relationships between variables and to solve systems of linear equations.

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