The problem asks us to find the function notation and equation form of a transformed absolute value function $f(x) = |x|$. The transformations are: a vertical stretch by a factor of 2, a vertical shift down by 4, and a horizontal shift right by 4.

1. Problem Description

The problem asks us to find the function notation and equation form of a transformed absolute value function $f(x) = |x|$. The transformations are: a vertical stretch by a factor of 2, a vertical shift down by 4, and a horizontal shift right by

4.

2. Solution Steps

First, consider the vertical stretch by a factor of

2. This transforms $f(x)$ into $2f(x)$.

Second, consider the vertical shift down by

4. This transforms $2f(x)$ into $2f(x) - 4$.

Third, consider the horizontal shift right by

4. This means replacing $x$ with $x-4$ in the function. So $2f(x) - 4$ becomes $2f(x-4) - 4$.

Therefore, the function notation is $g(x) = 2f(x-4) - 4$.

Since $f(x) = |x|$, we have $f(x-4) = |x-4|$. Thus, the equation form is $g(x) = 2|x-4| - 4$.

3. Final Answer

Function Notation: $g(x) = 2f(x-4) - 4$

Equation Form: $g(x) = 2|x-4| - 4$