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) is a non-vertical line in the plane. The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input. Linear functions are related to linear equations.
Properties A linear function is a polynomial function in which the variable x has degree at most one (a linear polynomial):
f ( x ) = a x + b {\displaystyle f(x)=ax+b} . Such a function is called linear because its graph, the set of all points ( x , f ( x ) ) {\displaystyle (x,f(x))} in the Cartesian plane, is a line. The coefficient a is called the slope of the function and of the line (see below). If the slope is a = 0 {\displaystyle a=0} , this is a constant function f ( x ) = b {\displaystyle f(x)=b} defining a horizontal line, which some authors exclude from the class of linear functions. With this definition, the degree of a linear polynomial would be exactly one, and its graph would be a line that is neither vertical nor horizontal. However, in this article, a ≠ 0 {\displaystyle a\neq 0} is not required, so constant functions will be considered linear. If b = 0 {\displaystyle b=0} then the linear function is said to be homogeneous. Such function defines a line that passes through the origin of the coordinate system, that is, the point ( x , y ) = ( 0 , 0 ) {\displaystyle (x,y)=(0,0)} . In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes b ≠ 0 {\displaystyle b\neq 0} . The natural domain of a linear function f ( x ) {\displaystyle f(x)} , the set of allowed input values for x, is the entire set of real numbers, x ∈ R . {\displaystyle x\in \mathbb {R} .} One can also consider such functions with x in an arbitrary field, taking the coefficients a, b in that field. The graph y = f ( x ) = a x + b {\displaystyle y=f(x)=ax+b} is a non-vertical line having exactly one intersection with the y-axis, its y-intercept point ( x , y ) = ( 0 , b ) . {\displaystyle (x,y)=(0,b).} The y-intercept value y = f ( 0 ) = b {\displaystyle y=f(0)=b} is also called the initial value of f ( x ) . {\displaystyle f(x).} If a ≠ 0 , {\displaystyle a\neq 0,} the graph is a non-horizontal line having exactly one intersection with the x-axis, the x-intercept point ( x , y ) = ( − b a , 0 ) . {\displaystyle (x,y)=(-{\tfrac {b}{a}},0).} The x-intercept value x = − b a , {\displaystyle x=-{\tfrac {b}{a}},} the solution of the equation f ( x ) = 0 , {\displaystyle f(x)=0,} is also called the root or zero of f ( x ) . {\displaystyle f(x).}
Slope
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