Linear function

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A linear function is a mathematical function of a certain form. Unfortunately, there is some disagreement as to exactly what that form is. A common definition, which is taught in many schools, is that a linear function is a function of the form:

f(x) = m x + c

where m and c are constants.

The problem with this 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 a function is of the above form with c equal to zero, the function does actually satisfy the properties of a linear map.

Linear functions (according to the first definition) can also be written in the form:

y = m x + c

and plotted on an x,y graph. It forms a straight line, as the name implies.

The constant m is often called the slope or gradient while c is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.


  • f(x)= 2x + 1

(here m=2, c=1)

  • f(x) = x

(m=1, c=0)

  • f(x)= 9 x - 2
  • f(x)= -3 x + 4

On a line graph, changing m makes the line steeper or shallower, and changing c moves the line up or down.

As mentioned, the line crosses the y-axis at the co-ordinate (0,c). It crosses the x-axis at (-c / m) (solving for 0 = m x + c we get x = -c / m).

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