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Decida cuál de las siguientes funciones tiene la gráfica dada.

To determine which function matches the given graph, we need to analyze the characteristics of the graph and match them with the corresponding function.

### Step-by-Step Analysis:
1. **Identify the roots (x-intercepts) of the graph:**
   - The graph crosses the x-axis at \( x = -2, 0, 1, \) and \( 2 \).

2. **Determine the behavior at each root:**
   - At \( x = -2 \), the graph touches the x-axis and turns around (indicating a double root).
   - At \( x = 0 \), \( x = 1 \), and \( x = 2 \), the graph crosses the x-axis.

3. **Match the characteristics with the provided functions:**

Let's analyze each function:

a) \( y = \frac{1}{2}(x^2 - 1)(x - 2) \):
   - Roots: \( x = \pm1, 2 \)
   - This function has roots at \( x = \pm1 \) and \( x = 2 \), which do not match our graph.

b) \( y = -\frac{1}{2}(x^2 + 1)(x - 2) \):
   - Roots: No real roots from \( x^2 + 1 \), and \( x = 2 \)
   - This function does not match our graph either.

c) \( y = (x^2 - 1)\left(1 - \frac{x}{2}\right) \):
   - Roots: \( x = \pm1, 2 \)
   - This function has roots at \( x = \pm1 \) and \( x = 2 \), which do not match our graph.

d) \( y = -\frac{1}{2}(x^2 - 1)^2(x - 2) \):
   - Roots: \( x = \pm1, 2 \)
   - This function has double roots at \( x = \pm1 \) and a root at \( x = 2 \), which do not match our graph.

e) \( y = \left(x^2 + \frac{1}{2}\right)(x^2 - 1)(2 - x) \):
   - Roots: \( x = \pm1, 2 \)
   - This function has roots at \( x = \pm1 \) and \( x = 2 \), which do not match our graph.

f) \( y = (x - 1)(x - 2)(x + 1) \):
   - Roots: \( x = 1, 2, -1 \)
   - This function has roots at \( x = 1, 2, \) and \( x = -1 \), which do not match our graph.

None of the provided functions have the correct roots matching \( x = -2, 0, 1, \) and \( 2 \). Therefore, let's re-evaluate the graph and the functions more carefully.

The correct function should match the roots \( x = -2, 0, 1, 2 \) and have the appropriate behavior at each intercept (touching or crossing). The graph indicates \( x = -2 \) is a double root, and \( x = 0, 1, 2 \) are simple roots.

After thorough examination, the given options do not correctly match the graph shown. There may be a mistake or misinterpretation in the provided options.

Would you like to explore this further, or do you have any other questions?

### Additional Questions:

1. How do you determine the multiplicity of roots from a graph?
2. What is the significance of the leading coefficient in a polynomial function?
3. How can the end behavior of a graph help identify the degree of a polynomial?
4. What is the importance of the y-intercept in graph analysis?
5. How do transformations affect the graph of a polynomial function?
6. Can you describe the difference between a double root and a simple root?
7. How do you factorize higher degree polynomials?
8. How does the derivative of a polynomial help in analyzing its graph?

### Tip:
When matching a graph to a function, always check the x-intercepts, their multiplicities, and the general shape of the graph for an accurate match.

Learn how to determine which function matches a given graph by analyzing its roots and behavior. This problem involves identifying x-intercepts, analyzing multiplicity of roots, and matching graph characteristics to polynomial functions. Suitable for high school students studying functions and graph analysis.

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