https://file.aiquickdraw.com/mathbot/file/1727847943224uk2ag3w9.jpg

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 \( x \).

The graph shows the function f(x) = |x| + 2. Place the appropriate symbols in the equation of g(x) so that g(x) is a transformation of f(x) that follows the rule of reflection across the x-axis.

Learn how to transform the function f(x) = |x| + 2 by reflecting it across the x-axis. This problem involves key algebraic concepts like absolute value functions and transformations. Suitable for students in Grades 8-10.

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