The answer is
![f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cbegin%7Bcases%7Dx-2%20%5Ctext%7B%20if%20%7Dx%20%5Cge%20-2%20%5C%5C%20-x-6%20%5Ctext%7B%20if%20%7Dx%20%3C%20-2%5Cend%7Bcases%7D)
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Here's how I got that answer:
Start with the piecewise definition for y = |x|.
![g(x) = \begin{cases}x \text{ if }x \ge 0 \\ -x \text{ if }x < 0\end{cases}](https://tex.z-dn.net/?f=g%28x%29%20%3D%20%5Cbegin%7Bcases%7Dx%20%5Ctext%7B%20if%20%7Dx%20%5Cge%200%20%5C%5C%20-x%20%5Ctext%7B%20if%20%7Dx%20%3C%200%5Cend%7Bcases%7D)
Everywhere you see an 'x', replace it with x+2
![g(x+2) = \begin{cases}x+2 \text{ if }x+2 \ge 0 \\ -(x+2) \text{ if }x+2 < 0\end{cases}](https://tex.z-dn.net/?f=g%28x%2B2%29%20%3D%20%5Cbegin%7Bcases%7Dx%2B2%20%20%5Ctext%7B%20if%20%7Dx%2B2%20%5Cge%200%20%5C%5C%20-%28x%2B2%29%20%5Ctext%7B%20if%20%7Dx%2B2%20%3C%200%5Cend%7Bcases%7D)
![g(x+2) = \begin{cases}x+2 \text{ if }x \ge -2 \\ -x-2 \text{ if }x < -2\end{cases}](https://tex.z-dn.net/?f=g%28x%2B2%29%20%3D%20%5Cbegin%7Bcases%7Dx%2B2%20%20%5Ctext%7B%20if%20%7Dx%20%5Cge%20-2%20%5C%5C%20-x-2%20%5Ctext%7B%20if%20%7Dx%20%3C%20-2%5Cend%7Bcases%7D)
Now tack on "-4" at the end of each piece so that we shift the function down 4 units
![g(x+2)-4 = \begin{cases}x+2-4 \text{ if }x \ge -2 \\ -x-2-4 \text{ if }x < -2\end{cases}](https://tex.z-dn.net/?f=g%28x%2B2%29-4%20%3D%20%5Cbegin%7Bcases%7Dx%2B2-4%20%20%5Ctext%7B%20if%20%7Dx%20%5Cge%20-2%20%5C%5C%20-x-2-4%20%5Ctext%7B%20if%20%7Dx%20%3C%20-2%5Cend%7Bcases%7D)
![g(x+2)-4 = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}](https://tex.z-dn.net/?f=g%28x%2B2%29-4%20%3D%20%5Cbegin%7Bcases%7Dx-2%20%20%5Ctext%7B%20if%20%7Dx%20%5Cge%20-2%20%5C%5C%20-x-6%20%5Ctext%7B%20if%20%7Dx%20%3C%20-2%5Cend%7Bcases%7D)
![f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cbegin%7Bcases%7Dx-2%20%20%5Ctext%7B%20if%20%7Dx%20%5Cge%20-2%20%5C%5C%20-x-6%20%5Ctext%7B%20if%20%7Dx%20%3C%20-2%5Cend%7Bcases%7D)
Check out the attached images below. In figure 1, I graph y = x-2 and y = -x-6 as separate equations on the same xy coordinate system. Then in figure 2, I combine them to form the familiar V shape you see with any absolute value graph.