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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!

The graph of the function y = g(x) represents a transformation of the graph of y = f(x). Determine the equation of g(x) in the form y = a f(b(x - h)) + k.

Learn how to determine the equation of the transformed function g(x) in the form y = a f(b(x - h)) + k. This problem covers the key transformations: vertical scaling, horizontal stretching, and translations, suitable for high school students.

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