Describe the transformation that takes f (x) = |x| to g (x) = - | x + 4 | - 1

Horizontal translation → Reflection about x-axis → Vertical Translation

Step-by-step explanation:

We are given the function f (x) = |x|.

This function is transformed to the new function given by g (x) = - |x+4| - 1.

So, we see that,

1. Translation of 4 units to the left is applied, which changes f (x) = |x| to |x+4|.

2. Reflection about x-axis is applied, which changes |x+4| to - |x+4|.

3. Finally, translation of 1 unit downward is applied, which changes - |x+4| to g (x) = - |x+4| - 1.

Hence, the sequence of transformation applied to obtain g (x) from f (x) is Horizontal translation → Reflection about x-axis → Vertical Translation.