Vertical translations occur outside the absolute value while horizontal translations occur inside the absolute value.
y=|x|+9
Practice makes perfect
We are asked to translate the equation y=|x| up by 9 units. Remember, when translating absolute value functions, vertical translations occur outside the absolute value. Consider the following general equation, where k is a real number.
y=|x|+ k
The graph of this equation is a vertical translation of y=|x| by k units. In this case, the value of k is 9. Translating up by 9 units means that we need to add this value.
y=|x| + 9
Extra
Horizontal and Vertical Translation
Let's recall what we know about horizontal and vertical translations of a graph of an absolute value function. Remember that a translation does not change the graph, it shifts the graph in vertical, horizontal or in both directions. Let's look at the graph of a function y=|x|.
We will start with the vertical translation. For an example, we would like to translate the graph of the function y=|x| 1 unit down. Keep in mind that vertical translation occurs outside the absolute value. When translating down we need to subtract the value.
y=|x| - 1
Let's take a look at the translated graph.
Next, we will focus on the horizontal translation. Let's see what happens when we want to translate a graph of y=|x|-1 3 units left. The most important thing to remember is that horizontal translation occurs inside the absolute value. When translating left we need to add the value.
y=|x + 3|-1
Now, let's take a look at the graph after two translations, the first one unit down and the second one, three units left.
