Vertical translations occur outside the absolute value while horizontal translations occur inside the absolute value.
Practice makes perfect
We are asked to graph the equation y=|x-4| by translating the graph of y=|x|. When translating absolute value functions, horizontal translations occur before evaluating the absolute value. Consider the following general equation, where h is a positive number.
y=|x- h|
The graph of this equation is a horizontal translation of the graph of y=|x| right h units. Let's start with the graph of the parent function y=|x|.
In the case of the given function, h is equal to 4. Therefore, the transformation will move the graph of y=|x| to the right by 4 units. As an example, we will translate three points on the graph of y=|x| to the right by 4 units.
Translating each and every point 4 units right means shifting the entire graph of y=|x| to the right by 4 units.
