Which quadratic function has a range that can be described as y≤10.5

Question

?(1 point)
Responses

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
f left parenthesis x right parenthesis equals 6 left parenthesis x plus 4 right parenthesis squared plus 10.5

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

Answer 1

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

1. \( f(x) = -6(x - 10.5)^2 + 4 \):
   This is a downward-opening parabola (since the coefficient of the squared term is negative). The maximum value (the vertex) occurs at \( y = 4 \), so the range is \( y \leq 4\).

2. \( f(x) = 6(x + 4)^2 + 10.5 \):
   This is an upward-opening parabola (positive coefficient). The minimum value (the vertex) occurs at \( y = 10.5\), so the range is \( y \geq 10.5 \).

3. \( f(x) = -6(x + 4)^2 + 10.5 \):
   This is also a downward-opening parabola. The maximum value (the vertex) occurs at \( y = 10.5\), so the range is \( y \leq 10.5\).

4. \( f(x) = 10.5(x - 4)^2 - 6 \):
   This is an upward-opening parabola. The minimum value occurs at \( y = -6\), so the range is \( y \geq -6\).

Given this analysis, the function that has a range described as \( y \leq 10.5 \) is:

\( f(x) = -6(x + 4)^2 + 10.5 \).