1. The problem is to write the absolute value function $g(x)$ in the form $a|x - h| + k$ given the vertex and the shape of the graph. 2. The vertex of the graph is given as $(-6, 2)$, which means $h = -6$ and $k = 2$ in the vertex form $a|x - h| + k$. 3. The graph opens upwards, so the coefficient $a$ is positive. Since the graph looks like the basic absolute value function shifted, we can assume $a = 1$ unless otherwise indicated. 4. Substitute the values into the vertex form: $$g(x) = 1|x - (-6)| + 2 = |x + 6| + 2$$ 5. This matches the vertex at $(-6, 2)$ and the graph opening upwards. Final answer: $$g(x) = |x + 6| + 2$$