Linear function

Linear function refers to a mathematical function that creates a straight line when graphed in a coordinate system. It is defined by the equation of the form y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept, which is the point at which the line crosses the y-axis. Linear functions are fundamental in mathematics and are used extensively across various fields, including statistics, physics, economics, and engineering, to model relationships where one variable changes at a constant rate with respect to another.

Definition

A linear function can be understood in the context of a two-dimensional space as any function that satisfies the following two properties:

• It is a polynomial function of degree at most one.
• Its graph in the Cartesian coordinate system is a straight line.

The general form of a linear function is given by:

y = mx + b

where:

• y is the value of the function at any given point of x,
• m is the slope of the line, which represents the rate of change of y with respect to x,
• x is the independent variable,
• b is the y-intercept, the point where the line crosses the y-axis.

Characteristics

Linear functions have several key characteristics:

• The slope (m) indicates the steepness of the line and the direction in which the line moves. A positive slope means the line ascends from left to right, while a negative slope means it descends.
• The y-intercept (b) provides a starting point for the line on the y-axis.
• The function is continuous and defined for all real numbers.
• The rate of change of the function is constant, which means the function increases or decreases at a steady rate.

Applications

Linear functions are used in various scientific and practical applications. Some examples include:

• In Economics, to model supply and demand curves.
• In Physics, to describe motion at a constant speed.
• In Statistics, for simple linear regression analysis to predict the value of a variable based on the value of another.
• In Engineering, to design structures and analyze forces.

Graphing Linear Functions

To graph a linear function, one needs to identify two main components from its equation: the slope (m) and the y-intercept (b). Starting at the y-intercept on the y-axis, the slope is used to determine the direction and steepness of the line. For example, a slope of 2 means that for every one unit increase in x, y increases by two units.

See Also

• Function (mathematics)
• Polynomial function
• Slope
• Y-intercept
• Coordinate system