The problem asks us to determine the equation of the function $g(x)$, which represents a transformation of the graph of $y = f(x)$. The transformed function should be expressed in the form $y = a f(b(x - h)) + k$.

Step-by-Step Explanation

1. Understanding the Transformations: The function $g(x)$ is a transformation of $f(x)$, which can involve:

   • Vertical stretching or shrinking (given by $a$).
   • Horizontal stretching or shrinking (given by $b$).
   • Horizontal translation (shift by $h$).
   • Vertical translation (shift by $k$).

2. Identify Key Points: From the graph, identify some key points of $f(x)$ and $g(x)$ to determine how the transformations have shifted the graph. The following points are visible:

   • $f(x)$ has a point at $(-4, 0)$ and at $(4, 0)$ for its roots, and a peak at $(0, 4)$.
   • $g(x)$ has a point at $(-6, 0)$, another at $(6, 0)$, and a peak at $(0, 2)$.

3. Determine Horizontal Shift ( $h$ ): The graph of $g(x)$ appears to be a horizontal shift of $f(x)$. We observe that:

   • The zeroes of $f(x)$ are at $x = -4$ and $x = 4$.
   • The zeroes of $g(x)$ are at $x = -6$ and $x = 6$.

   This indicates that the graph of $f(x)$ has been shifted horizontally by 2 units to the left. Therefore, $h = -2$.

4. Determine Vertical Scaling ( $a$ ): The peak of $f(x)$ is at $(0, 4)$, while the peak of $g(x)$ is at $(0, 2)$. This shows that the graph has been vertically shrunk by a factor of 2. Thus, $a = \frac{1}{2}$.

5. Determine Horizontal Scaling ( $b$ ): The distance between the roots of $f(x)$ is $4 - (-4) = 8$, and for $g(x)$, it is $6 - (-6) = 12$. This indicates a horizontal stretching by a factor of $\frac{3}{2}$. Thus, $b = \frac{2}{3}$.

6. Determine Vertical Shift ( $k$ ): There is no vertical shift since both $f(x)$ and $g(x)$ have their roots on the x-axis. Thus, $k = 0$.

Final Equation

Based on these transformations, the equation of $g(x)$ is: $g(x) = \frac{1}{2} f\left( \frac{2}{3}(x + 2) \right)$

Would you like more details or clarifications on any part of the explanation?

Here are 5 related questions for practice:

1. What would happen if the vertical stretch factor $a$ was negative?
2. How does the horizontal stretch factor $b$ affect the graph when $b > 1$ versus $0 < b < 1$?
3. What is the effect of changing $k$ on the graph?
4. How would the graph change if $h$ was positive instead of negative?
5. Can you derive the transformation equation if the horizontal stretch was $\frac{1}{2}$?

Tip: Always verify your transformations by comparing key points from both the original and transformed graphs to ensure accuracy!