Question

Which quadratic function has a range that can be described as y≤10.5 ?(1 point) Responses f(x)=−6(x+4)2+10.5 f left parenthesis x right parenthesis equals negative 6 left parenthesis x plus 4 right parenthesis squared plus 10.5 f(x)=10.5(x−4)2−6 f left parenthesis x right parenthesis equals 10.5 left parenthesis x minus 4 right parenthesis squared minus 6 f(x)=−6(x−10.5)2+4 f left parenthesis x right parenthesis equals negative 6 left parenthesis x minus 10.5 right parenthesis squared plus 4 f(x)=6(x+4)2+10.5

Answer 1

To determine which quadratic function has a range described as \( y \leq 10.5 \), we need to analyze the vertex and the direction of opening for each function.

The general form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h,k) \) is the vertex. The value of \( k \) represents the maximum or minimum value of the quadratic function depending on whether the parabola opens upward or downward:

- If \( a < 0 \) (opens downwards), the maximum value of the function occurs at \( k \).
- If \( a > 0 \) (opens upwards), the minimum value of the function occurs at \( k \).

Given the range \( y \leq 10.5 \), the function must open downwards, indicating that the coefficient \( a \) must be negative. Therefore, we will look for a function with a vertex where \( k = 10.5 \) and \( a < 0 \).

Let's evaluate the options:

1. **\( f(x) = -6(x+4)^2 + 10.5 \)**:
- The vertex is \( (−4, 10.5) \).
- Since \( a = -6 < 0 \), the range is \( y \leq 10.5 \).

2. **\( f(x) = 10.5(x−4)^2 − 6 \)**:
- The vertex is \( (4, -6) \).
- Since \( a = 10.5 > 0 \), the range is \( y \geq -6 \).

3. **\( f(x) = -6(x−10.5)^2 + 4 \)**:
- The vertex is \( (10.5, 4) \).
- Since \( a = -6 < 0 \), the range is \( y \leq 4 \).

4. **\( f(x) = -6(x−4)^2 + 10.5 \)**:
- The vertex is \( (4, 10.5) \).
- Since \( a = -6 < 0 \), the range is \( y \leq 10.5 \).

5. **\( f(x) = 6(x+4)^2 + 10.5 \)**:
- The vertex is \( (−4, 10.5) \).
- Since \( a = 6 > 0 \), the range is \( y \geq 10.5 \).

The functions that satisfy the condition \( y \leq 10.5 \) are:
- \( f(x) = -6(x+4)^2 + 10.5 \)
- \( f(x) = -6(x−4)^2 + 10.5 \)

Both options 1 and 4 have the desired range. However, since the question only asks for one function, either option can be selected as the correct response, often depending on the context of previous exercises - but based on the phrasing of the question, I will go with **\( f(x) = -6(x+4)^2 + 10.5 \)**.