Question

What are the coordinates of the minimum point for the square root function that relates to the quadratic function f(x)=−3(x−5)^2 +7 (2 points) (5,7), or (7,5). explain your answer

Answer 1

To find the coordinates of the minimum point of the quadratic function \( f(x) = -3(x - 5)^2 + 7 \), we first observe the structure of the function.

1. **Identify the vertex form**: The function is in the vertex form of a quadratic equation, which is given by \( f(x) = a(x - h)^2 + k \). In this format:
- \( a = -3 \)
- \( h = 5 \)
- \( k = 7 \)

2. **Determine the vertex**: The vertex (h, k) of the quadratic function is the point which represents either the maximum or minimum of the function. Because the coefficient \( a = -3 \) is negative, the parabola opens downward, meaning that the vertex is the maximum point.

3. **Coordinates of the vertex**: From our values of \( h \) and \( k \), the coordinates of the vertex are \( (h, k) = (5, 7) \).

4. **Minimum Point Explanation**: In this case, since the function opens downward, there is no minimum point; instead, the maximum point at the vertex is \( (5, 7) \).

Thus, evaluating the given choices:
- **(5, 7)** is the vertex and represents the maximum point.
- **(7, 5)** does not correspond to any valid point of interest in this function.

Thus, the answer is **(5, 7)**.