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Linear function
From Wikipedia, the free encyclopedia
The term linear function can refer to either of two slightly different concepts. In geometry and elementary algebra a linear function is a first degree polynomial (generally a straight line) mathematical function of the form:
- f(x) = mx + b
where m and b are real constants and x is a real variable.
The problem with this geometric definition is that functions of the above form -despite their names- do not necessarily satisfy the conditions of a linear map. Therefore, some people refer to functions of the above form as affine functions. If and only if a function is of the above form with b equal to zero does the function satisfy the properties of a linear map, preserving scalar multiplication and vector addition for all points in its domain.
The constant m is often called the slope or gradient while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.
Examples:
- f(x)= 2x + 1
- f(x) = x
- f(x)= 9 x − 2
- f(x)= −3 x + 4
On a line graph, changing m makes the line steeper or shallower, and changing b moves the line up or down.
As mentioned, the line crosses the y-axis at the coordinate (0, b). If m ≠ 0, it crosses the x-axis at (−b⁄m).