# Linear function

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A linear function can refer to 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) = m x + 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 co-ordinate (0,b). If m≠0, it crosses the x-axis at (-b / m).