https://file.aiquickdraw.com/mathbot/file/172905120868634aa9kwm.jpg

Please look at the table/graph, and answer the fill in the blanks on the right

Let's analyze the graph and table to answer the questions.

1. **Roots of the function:**  
   The roots are the values of \(x\) where \(f(x) = 0\). From the table and graph, the function \(f(x) = x^2 - x - 6\) is zero at:
   - \(x = -2\)
   - \(x = 3\)

2. **Vertex of the function:**  
   The vertex is the minimum point of the parabola since the parabola opens upwards (as indicated by the positive coefficient of \(x^2\)). By observing the graph, the vertex seems to be at \(x = 0.5\), with a corresponding value \(f(0.5) \approx -6.25\). This is an estimate based on the curvature and nearby points provided in the table.

### Answers for the blanks:

- **Where are the roots of the function according to the graph?**  
  **The roots are at \(x = -2\) and \(x = 3\).**

- **Where is the vertex of the function according to the graph? (Give your best estimate!)**  
  **The vertex is approximately at \((0.5, -6.25)\).**

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Would you like further details or clarifications?

Here are 5 follow-up questions related to the problem:
1. How can you find the exact vertex using the formula for parabolas?
2. Can you derive the roots algebraically for this quadratic equation?
3. How does changing the coefficient of \(x^2\) affect the shape of the graph?
4. What is the axis of symmetry of the function \(f(x) = x^2 - x - 6\)?
5. How does the vertex relate to the minimum or maximum value of the function?

**Tip:** To find the vertex of a quadratic function \(f(x) = ax^2 + bx + c\), use the formula for the x-coordinate: \(x = -\frac{b}{2a}\).

Please look at the table/graph, and answer the fill in the blanks on the right.

Analyze the quadratic function x^2 - x - 6. Learn how to determine its roots at x = -2 and x = 3, and estimate the vertex at approximately (0.5, -6.25). Suitable for Grades 9-12.

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