Given the function \( F(x) = -3(x - 2)^2 + 6 \), let's analyze it in detail.

### 1. **Vertex Form**:
The function is already in vertex form, \( F(x) = a(x - h)^2 + k \), where:
- \( a = -3 \) (which determines the direction and width of the parabola)
- \( h = 2 \) (the x-coordinate of the vertex)
- \( k = 6 \) (the y-coordinate of the vertex)

### 2. **Vertex**:
The vertex of the parabola is at \( (h, k) = (2, 6) \). This is the highest point on the graph since the parabola opens downwards (because \( a < 0 \)).

### 3. **Direction and Width**:
- The parabola opens **downwards** because \( a = -3 \) is negative.
- The factor \( |a| = 3 \) indicates the parabola is narrower than the standard parabola \( y = x^2 \).

### 4. **Axis of Symmetry**:
The axis of symmetry is the vertical line that passes through the vertex, given by \( x = h \), so:
\[
x = 2
\]

### 5. **Y-intercept**:
To find the y-intercept, set \( x = 0 \) in the function:
\[
F(0) = -3(0 - 2)^2 + 6 = -3(4) + 6 = -12 + 6 = -6
\]
So, the y-intercept is \( (0, -6) \).

### 6. **X-intercepts**:
To find the x-intercepts, set \( F(x) = 0 \) and solve for \( x \):
\[
0 = -3(x - 2)^2 + 6
\]
Subtract 6 from both sides:
\[
-6 = -3(x - 2)^2
\]
Divide both sides by -3:
\[
2 = (x - 2)^2
\]
Take the square root of both sides:
\[
x - 2 = \pm \sqrt{2}
\]
So the solutions are:
\[
x = 2 + \sqrt{2} \quad \text{and} \quad x = 2 - \sqrt{2}
\]
These are the x-intercepts.

### Summary:
- Vertex: \( (2, 6) \)
- Axis of Symmetry: \( x = 2 \)
- Y-intercept: \( (0, -6) \)
- X-intercepts: \( x = 2 + \sqrt{2}, 2 - \sqrt{2} \)
- The parabola opens downward and is narrower than the standard parabola.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How does changing the value of \( a \) in the vertex form \( a(x-h)^2+k \) affect the graph?
2. How do you find the x-intercepts of a quadratic function using the quadratic formula?
3. What is the importance of the axis of symmetry in the graph of a quadratic function?
4. How would the graph change if the vertex of the parabola was at the origin \( (0,0) \)?
5. How do transformations of the graph of \( y = x^2 \) correspond to changes in the vertex form equation?

**Tip:** Always check the sign of \( a \) in the vertex form to determine the direction in which the parabola opens.

Explore the function F(x)=-3(x-2)^2+6 in detail, including its vertex, axis of symmetry, y-intercept, x-intercepts, and direction of opening. This analysis covers key concepts in quadratic functions and vertex form equations. Suitable for Grades 9-12.

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