To write an equation that represents a translation 1 unit down and 5 units right of y = |x|, you would modify the equation as follows: y = |x - 5| - 1

To create the equation for a translation of the graph of y = |x|, we have to understand what happens when we translate a graph in the Cartesian coordinate system. A translation involves sliding the entire graph of a function in either the horizontal or vertical direction (or both).

For a horizontal translation (left or right), you modify the x-variable. Moving the graph to the right by "h" units will replace x with (x - h) in the equation. If you move it left by "h" units, you would replace x with (x + h).

For a vertical translation (up or down), you modify the whole function. Moving the graph up by "k" units adds "k" to the function's output. Moving it down by "k" units subtracts "k".

To translate y = |x| 1 unit down, we subtract 1 from the function, resulting in y - 1 = |x|. To translate it 5 units right, we replace x with (x - 5), resulting in y = |x - 5|.

Combining these two translations, the equation becomes y = |x - 5| - 1. This represents the graph of y = |x| translated 1 unit down and 5 units right.