the actual answer was f(x) = (3/2)(x-4)^2(x+5)^2

Thank you for sharing the correct answer. Let's analyze how the function \( f(x) = \frac{3}{2}(x - 4)^2(x + 5)^2 \) fits the graph you provided.

### Breakdown of the Function:

1. **Form of the Function:**
   - The function \( f(x) = \frac{3}{2}(x - 4)^2(x + 5)^2 \) is a quartic polynomial because the highest degree term would be \( x^4 \) after expanding the factors.
   - The factors \( (x - 4)^2 \) and \( (x + 5)^2 \) indicate that the graph has double roots at \( x = 4 \) and \( x = -5 \).

2. **Graph Characteristics:**
   - **Double Roots:** The double roots at \( x = 4 \) and \( x = -5 \) suggest that the graph touches the x-axis at these points and does not cross it, which is consistent with the squared terms.
   - **End Behavior:** Since the coefficient \(\frac{3}{2}\) is positive and the highest degree term is \( x^4 \), as \( x \to \pm\infty \), \( f(x) \to \infty \). This matches the end behavior seen in the graph.
   - **Vertex/Turning Points:** The graph has turning points at \( x = 4 \) and \( x = -5 \), consistent with the points where the function touches the x-axis.

### Comparison to the Graph:
- **Touching the X-axis:** The graph touches the x-axis at \( x = 4 \) and \( x = -5 \), corresponding to the zeros of the function.
- **Shape:** The graph exhibits the correct quartic behavior with two minima corresponding to the double roots and no other x-intercepts.

### Summary:
The function \( f(x) = \frac{3}{2}(x - 4)^2(x + 5)^2 \) perfectly describes a graph that touches the x-axis at \( x = 4 \) and \( x = -5 \), with the ends of the graph rising to infinity. This indicates that it fits the graph shown in the image very well.

Would you like more details on how this function is derived or have any further questions?

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Here are 5 related questions to explore further:

1. How would the graph of \( f(x) = \frac{3}{2}(x - 4)^2(x + 5)^2 \) change if the coefficients or powers of the factors were altered?
2. What are the general properties of graphs of polynomial functions with even degree and positive leading coefficients?
3. How does the multiplicity of roots affect the shape of a polynomial graph at the x-intercepts?
4. What methods can you use to find the roots of a polynomial function from its equation?
5. How would the graph be different if the function had a term like \( (x-2) \) instead of \( (x-4)^2 \)?

**Tip:** When analyzing polynomial functions, pay close attention to the multiplicity of the roots. Roots with even multiplicity indicate points where the graph touches but does not cross the x-axis.

Explore the function f(x) = (3/2)(x - 4)^2(x + 5)^2 in depth with detailed graph analysis and mathematical breakdown. This problem involves polynomial functions and quadratic equations, examining graph characteristics such as double roots and end behavior. Suitable for advanced high school students and beyond.

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