In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero).
When the function is of only one variable, it is of the form
where a and b are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept.
For a function of any finite number of independent variables, the general formula is
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one independent variable, is a horizontal line.
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates with uniform scales) is a line in the plane. Their characteristic property that when the value of the input variable is changed, the change in the output is a constant multiple of the change in the input variable.
A linear function is a polynomial function in which the variablex has degree at most one, which means it is of the form
Here x is the variable. The graph of a linear function, that is, the set of all points whose coordinates have the form (x, f(x)), is a line on the Cartesian plane (if over real numbers). That is why this type of function is called linear. Some authors, for various reasons, also require that the coefficient of the variable (the a in ax + b) should not be zero. This requirement can also be expressed by saying that the degree of the polynomial defining the function is exactly one, or by saying that the line which is the graph of a linear function is a slanted line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, f(x) = b, will be considered to be linear functions (their graphs are horizontal lines).
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of any finite number of 
