An absolute function is a function that is always positive.

f(x) = | x | ± k where k moves the graph for the function f(x) = |x| upwards by ‘k’ if k is positive, and it moves the graph of the function down by ‘k’ if k is negative.

f(-x) = | -x | ± k

= | x | ± k

= f(x)

An absolute function is always an even function.

If the absolute function is given by

f(x) = | x ± k | where k moves the graph f(x) = |x| to the right by ‘k’ if k is negative, and the graph will move to the left by ‘k’ if k is positive. In other words

f(x) = {x if x ≥ 0

{-x if x < 0

To graph an absolute function, it is advisable to first make a table of values.

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

f(x) | 3 | 2 | 1 | 0 | 1 | 2 | 3 |

Plotting these values we get the graph of the absolute function | x | ± k

This graph is a combination of the lines

y = x x ≥ 0 and

y = -x x < 0

Translate y = | x | according to k for y = | x ± k | and y = | x | ± k as explained above.

## Examples of graphing absolute functions:

**Example 1**: Draw y = f(x) = | x | – 1.

To get this graph, follow all the steps to draw the graph for y = f(x) = |x|. Then simply move the graph down by 1 unit along the y-axis.

x-intercepts: y = 0; 0 = |x| -1, |x| = 1, or x = ± 1

y-intercepts: x = 0; y = -1 = f(0).

Domain: {all real x}

Range: {y: y ≥ -1}

**Example 2**: Draw y = f(x) = |x| + 1.

x-intercepts: y = 0; 0 = |x| = -1 is impossible, so the graph will never meet the x-axis

y-intercepts: x = 0; y = +1

Domain: {all real x}

Range: {y: y ≥ 1}

**Example 3**: Draw y = f(x) = |x + 1|

x-intercepts: y = 0; 0 = |x + 1| = 0 = x + 1 = 0 or x = -1 (-1, 0)

y-intercepts: x = 0; y = +1 (0, 1)

This graph is f(x) = |x| moved by 1 unit to the left

Domain: {all real x}

Range: {y: y ≥ 0}

**Example 4**: Draw y = f(x) = |x – 1|

x-intercepts: y = 0; 0 = |x – 1| = 0

= x – 1 = 0 or x = 1 (1, 0)

y-intercepts: x = 0; y = |-1| = (0, 1)

This graph is f(x) = |x| moved by 1 unit to the right

Domain: {all real x}

Range: {y: y ≥ 0}

## Graphing Absolute Functions : Summary

For an absolute function y = f(x) = |x|,

Domain: {all real x}

Range: {y: y ≥ 0}

Its graph is ‘V’ shaped and passing through the intercepts.