The graphs of $f$ and $g$ appear to be transformations of each other. To determine the equation of $g(x)$, let’s analyze the relationship between $f(x)$ and $g(x)$:

1. The function $f(x)$ is given by: $f(x) = \frac{1}{4}(x + 2)^3 - 2$

2. Observing the graph:

   • The graph of $g(x)$ appears to be a reflection of $f(x)$ over the $x$-axis.
   • A reflection over the $x$-axis changes the sign of the output of $f(x)$, giving $g(x) = -f(x)$.

3. Applying the reflection: Substituting $f(x)$ into $g(x)$, we have: $g(x) = -f(x) = -\left(\frac{1}{4}(x + 2)^3 - 2\right)$

4. Simplify: Distribute the negative sign: $g(x) = -\frac{1}{4}(x + 2)^3 + 2$

Thus, the equation of $g(x)$ is: $g(x) = -\frac{1}{4}(x + 2)^3 + 2$

Summary:

The equation of $g(x)$ is: $g(x) = -\frac{1}{4}(x + 2)^3 + 2$

Would you like me to plot or verify this result?

Relative Questions:

1. How do reflections over the $x$- and $y$-axes affect equations of functions?
2. What other transformations could make $g(x)$ from $f(x)$?
3. How does the vertex of $f(x)$ and $g(x)$ relate geometrically?
4. Can $g(x)$ be represented as a shifted version of another cubic function?
5. How would $g(x)$ behave if reflected again?

Tip:

When reflecting a graph over the $x$-axis, negate the entire function $f(x)$.