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To determine the range of the function \( f(x) = -x^2 + bx + c \) given the graph and the details provided, let's analyze the key information:

### Given:
1. The graph of \( f(x) = -x^2 + bx + c \) is a parabola that opens downwards (since the coefficient of \( x^2 \) is negative).
2. The turning point \( A \) is the vertex of the parabola.
3. The axis of symmetry is \( x = 2 \), so the x-coordinate of the vertex (turning point) \( A \) is \( 2 \).
4. The parabola intersects the x-axis at points \( B(-5, 0) \) and \( C \).

### Step 1: Find the y-coordinate of the vertex
Since the axis of symmetry is \( x = 2 \), we can substitute \( x = 2 \) into the equation \( f(x) \) to find the y-coordinate of the vertex. The y-coordinate of the vertex represents the maximum value of the function because the parabola opens downwards.

However, to find the exact value of \( f(2) \), we would need to determine the coefficients \( b \) and \( c \), which would require solving the system of equations derived from the given points \( B(-5, 0) \) and \( C \).

### Step 2: Analyze the range
Given that the parabola opens downward, the maximum value of \( f(x) \) is at the vertex \( A(2, f(2)) \), and the range of the function \( f(x) \) would be all values \( y \leq f(2) \).

To find \( f(2) \), the maximum value, we could proceed by:
1. Using the fact that \( f(-5) = 0 \).
2. Expressing \( f(x) = -x^2 + bx + c \) and using the points \( B(-5, 0) \) and \( C(x_C, 0) \) (where \( x_C \) is another root).

### Step 3: Conclusion
Without the exact values of \( b \) and \( c \), we can only state the general form of the range as:
\[ \text{Range of } f(x) = (-\infty, f(2)] \]

To determine the precise value of \( f(2) \), we would need more information about the specific points \( B \) and \( C \). However, based on the graph, if it were determined or given that \( f(2) = k \) for some maximum \( k \), then the range would be \( (-\infty, k] \).

Would you like to go into more details on how to find \( f(2) \) or explore any other related aspects?

### Related Questions:
1. How can the coefficients \( b \) and \( c \) be determined from the given points on the parabola?
2. What is the importance of the axis of symmetry in determining the vertex of a parabola?
3. How does the sign of the coefficient of \( x^2 \) affect the range of a quadratic function?
4. What methods can be used to find the roots of a quadratic equation?
5. How does the distance between the roots of a quadratic function affect its graph?

### Tip:
When analyzing the range of a quadratic function, always consider the direction of the parabola (upwards or downwards) and the value at its vertex, as this will guide you to the correct range.

Explore how to determine the range of the quadratic function f(x) = -x^2 + bx + c, given key points and graph details. Learn about vertex form, axis of symmetry, and the impact of the parabola's opening direction. Suitable for Grades 10-12.

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