https://file.aiquickdraw.com/mathbot/file/1728974452983g8o638qz.jpg

To analyze the given problem, let's observe the quadratic function graph provided in the image.

### Analyzing the Information:
- The graph has its vertex at the point \((1, 6)\). Since the parabola opens downward, the leading coefficient of the quadratic function must be negative.
- The x-intercepts appear to be at \(-2\) and \(3\).

### Vertex Form of a Quadratic Function:
The vertex form of a quadratic function is:
\[
f(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola. Substituting \((h, k) = (1, 6)\), the equation becomes:
\[
f(x) = a(x - 1)^2 + 6
\]

### Finding the Correct Equation:
To find the correct value of \(a\), we can use the x-intercepts (\(-2\) and \(3\)) to form the equation. Expanding this information into the standard form of a quadratic will lead us to:
\[
f(x) = -x^2 + 2x + 6
\]
Based on the available options, let's compare this with each given option.

### Given Options:
A. \(f(x) = 6 - x - x^2\)  
B. \(f(x) = -6 + x + x^2\)  
C. \(f(x) = -6 + x + x^2\)  
D. \(f(x) = 6 + x - x^2\)  

The correct choice is \(f(x) = 6 + x - x^2\), which matches the characteristics of the given graph.

### Final Answer:
**The correct answer is option D: \(f(x) = 6 + x - x^2\).**

Would you like more details or have any questions on how this solution was reached?

### Related Questions:
1. How do you convert a quadratic function from vertex form to standard form?
2. What are the steps to find the vertex of a quadratic function using its equation?
3. How can you determine the direction of a parabola from its equation?
4. How do you calculate the x-intercepts of a quadratic equation?
5. What role does the leading coefficient play in the shape of a parabola?

### Tip:
Always check the sign of the leading coefficient in a quadratic equation to determine the direction in which the parabola opens.

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Learn how to determine the quadratic function based on a graph with vertex (1,6) and x-intercepts at -2 and 3. This example demonstrates key concepts like vertex form and standard form of a quadratic equation.

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