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Linear Functions

A linear function is a straight line graph. Any straight line is given by one of two forms:

  1. Gradient form: y = mx + c, where m is the gradient/slope of the function and c is the y-intercept
  2. General form: ax + by + c = 0. This function has domain {all real x} and the range {all real y} where a and b are non-zero.



Let y = f(x) = 2x + 1.

First, we have to find the x-intercept when y = 0.

So 2x + 1 = 0, gives x = –1/2

Next we have to find the y-intercept when x = 0.

So y = 2×0 + 1 = 1.

Thus the x and y intercepts are (-1/2, 0) and (0, 1).

Now we need to find out if the function is odd or even.

f(x) = 2x + 1;  f(-x) = 2 x (-x) + 1 = -2x + 1.

This function is neither odd nor even.

With these two intercepts viz. (-1/2, 0) and (0, 1), we can now draw the straight line.

graph of a linear function

Domain: { all real x} and Range: { all real y}


There are two special types of linear functions, viz. horizontal lines and vertical lines.

Horizontal line:

When x = 0 in ax + by + c = 0, we get by + c = 0, or

y = –c/b

This is a constant function that is parallel to the x-axis, and will be a horizontal line passing through –c/b at y-axis. This type of function is called a horizontal line or constant horizontal linear function.

Vertical & Horizontal line

The domain and range for horizontal linear function are:

Domain: {all real x}

Range: {y: y = –c/b}, i.e. y is always –c/b


Vertical line:

When y = 0 in ax + by + c = 0, we get ax = -c, or

x = –c/a

This constant function is parallel to the y-axis, and will be a vertical line passing through –c/a at x-axis. This type of function is called a vertical line or constant vertical linear function.

The domain and range for vertical linear function are:

Domain: {x: x = –c/a}, i.e. x is always –c/a

Range: {all real y}


In summary

x = a is a vertical line with x-intercept ‘a’

Domain: {x: x = a}

Range: {all real y}

y = b is a vertical line with y-intercept ‘b’

Domain: {all real x}

Range: {y: y = b}



  • A linear function will be odd if it passes through the origin (y-intercept = 0)
  • A linear function will be even if even if it has only a constant (vertical lines only)
  • A general linear function is neither odd or even