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Answer the questions below based on the graph of the quadratic function f, of, xf(x) and a table of selected values of the quadratic function g, of, x, .g(x)

Thank you for sharing the graph and table. I will go through the details of each function \( f(x) \) and \( g(x) \) based on the provided information. Here’s what we can infer from each component:

### Observations about the Graph of \( f(x) \):
1. The function \( f(x) \) shown in the graph appears to be a **quadratic function** due to its parabolic shape.
2. The parabola opens **upward**, indicating that the leading coefficient of the quadratic function is positive.
3. The vertex of \( f(x) \) seems to be located at approximately \( (-1, -4) \), which suggests that the minimum value of \( f(x) \) occurs at \( x = -1 \).
4. The axis of symmetry for the graph is likely \( x = -1 \).
5. The function appears symmetric about this axis, as expected for a quadratic function.

### Observations about the Table for \( g(x) \):
The table provides values of \( g(x) \) for specific \( x \)-values. We can analyze these values to understand the function \( g(x) \).

| \( x \) | \( g(x) \) |
|---------|------------|
| -3      | 35        |
| -2      | 21        |
| -1      | 11        |
| 0       | 5         |
| 1       | 3         |
| 2       | 5         |
| 3       | 11        |

1. The values in the table suggest that \( g(x) \) also represents a **quadratic function** because the outputs (values of \( g(x) \)) follow a parabolic trend.
2. The smallest value of \( g(x) \) occurs at \( x = 1 \), with \( g(1) = 3 \). This indicates that the vertex of \( g(x) \) is likely at \( x = 1 \).
3. The values are symmetric around \( x = 1 \), which is a common feature of quadratic functions centered at their vertex.
4. Based on the table, \( g(x) \) also appears to be an upward-opening parabola, similar to \( f(x) \).

### Questions
Based on the graph and table, please answer the following questions:

1. **What is the minimum value of \( f(x) \) according to the graph?**
2. **What is the vertex of the function \( g(x) \) based on the table?**
3. **Do \( f(x) \) and \( g(x) \) have the same vertex?**
4. **At which points do \( f(x) \) and \( g(x) \) share the same \( x \)-values for corresponding \( y \)-values?**
5. **Is the symmetry of \( f(x) \) and \( g(x) \) about their respective vertices consistent?**

### Tip:
When analyzing quadratic functions, identifying the vertex is crucial, as it reveals the function's maximum or minimum point and helps understand the symmetry of the function.

Answer the questions below based on the graph of the quadratic function \( f(x) \) and a table of selected values of the quadratic function \( g(x) \).

Analyze and answer questions about the quadratic functions \( f(x) \) and \( g(x) \) using a graph and table of values. Explore symmetry, vertices, and minimum values in this problem, suitable for Grades 9-11.

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