Determining the function with vertex (2, -9): Analysis of Quadratic Functions.

Which function has a vertex at (2, -9)?f(x) = -(x – 3)2f(x) = (x + 8)2f(x) = (x – 5)(x + 1)f(x) = -(x – 1)(x – 5)

C f(x) = (x – 5)(x + 1)

To determine which function has a vertex at (2, -9), we need to analyze the vertex form of the quadratic function, which is represented as f(x) = a(x – h)² + k, where (h, k) represents the coordinates of the vertex.

Let’s analyze each given function to see if it matches the vertex (2, -9).

1. f(x) = -(x – 3)²:
In this function, h = 3 is the x-coordinate of the vertex, and k = 0 since there is no constant term added. Thus, the vertex is at (3, 0), not (2, -9). This function does not have the desired vertex.

2. f(x) = (x + 8)²:
In this function, h = -8 is the x-coordinate of the vertex, and k = 0 since there is no constant term added. The vertex is at (-8, 0), not (2, -9). Hence, this function does not have the required vertex.

3. f(x) = (x – 5)(x + 1):
This is a quadratic function in factored form, f(x) = x² – 4x – 5. To find the coordinates of the vertex, we need to use the formula -b/2a. In this case, a = 1 and b = -4. Thus, x = -(-4)/(2*1) = 2. Since x = 2, we can substitute it into the function to find the corresponding y-coordinate: f(2) = 2² – 4(2) – 5 = -1. The vertex of this function is (2, -1), not (2, -9).

4. f(x) = -(x – 1)(x – 5):
Similar to the previous function, this is also a quadratic function in factored form. Expanding it, we have f(x) = -x² + 6x – 5. Using the vertex formula as before, x = -2b/2a = -6/(-2) = 3. When we substitute x = 3 into the function, we find f(3) = -9. Thus, the vertex is (3, -9), which matches the desired vertex (2, -9).

In conclusion, the correct function that has a vertex at (2, -9) is f(x) = -(x – 1)(x – 5).

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