To solve this problem, we need to determine the equation for $g(x)$, which is a transformation of $f(x) = |x| + 2$.

From the graph, $f(x)$ has a V-shape with its vertex at $(0, 2)$ and represents the equation $f(x) = |x| + 2$. The function $g(x)$ appears to be a reflection of $f(x)$ across the x-axis and shifted downward.

The rule that follows a reflection across the x-axis involves multiplying the function by $-1$. Thus, the transformation of $f(x)$ that reflects it across the x-axis is:

$g(x) = -f(x)$

Substituting $f(x) = |x| + 2$, we get:

$g(x) = -(|x| + 2)$

Simplifying:

$g(x) = -|x| - 2$

This equation matches the graph, where $g(x)$ has its vertex at $(0, -2)$ and opens downward.

Final Answer:

$g(x) = -|x| - 2$

Do you need further details or have any other questions?

Here are 5 related questions to further explore this concept:

1. How would you transform $f(x) = |x| + 2$ to reflect it across the y-axis?
2. What would the equation for $f(x)$ look like if it were shifted upward by 3 units?
3. How does reflecting a function across the y-axis affect its equation?
4. If you were to stretch the graph of $f(x)$ vertically, how would the equation change?
5. How can we identify the vertex of an absolute value function based on its equation?

Tip: When transforming graphs of functions, always remember that reflections across the x-axis multiply the function by $-1$, while reflections across the y-axis alter the input by negating $x$.