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.

Exercise