Question
Which quadratic function has a range that can be described as y≤10.5
?(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
?(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
Answers
GPT-4o mini
To determine which quadratic function has a range expressed as \( y \leq 10.5 \), we need to identify the functions that open downward (since the maximum value would then be at the vertex). This occurs when the coefficient of the squared term is negative.
Let's analyze the given functions:
1. **\( f(x) = -6(x - 10.5)^2 + 4 \)**
This function opens downward (since the coefficient of the squared term is -6). The vertex is at \( (10.5, 4) \), so the maximum value of this function is 4. This does not satisfy \( y \leq 10.5 \).
2. **\( f(x) = 6(x + 4)^2 + 10.5 \)**
This function opens upward (since the coefficient of the squared term is +6). Thus, the minimum value occurs at the vertex \( (−4, 10.5) \). This does not satisfy \( y \leq 10.5 \).
3. **\( f(x) = -6(x + 4)^2 + 10.5 \)**
This function opens downward (since the coefficient of the squared term is -6). The vertex is at \( (−4, 10.5) \), and the maximum value is 10.5, leading to the range \( y \leq 10.5 \). This satisfies the condition.
4. **\( f(x) = 10.5(x - 4)^2 - 6 \)**
This function opens upward (since the coefficient of the squared term is +10.5). Therefore, the minimum value occurs and this does not satisfy \( y \leq 10.5 \).
Thus, the quadratic function with the range \( y \leq 10.5 \) is:
**\( f(x) = -6(x + 4)^2 + 10.5 \)**
Let's analyze the given functions:
1. **\( f(x) = -6(x - 10.5)^2 + 4 \)**
This function opens downward (since the coefficient of the squared term is -6). The vertex is at \( (10.5, 4) \), so the maximum value of this function is 4. This does not satisfy \( y \leq 10.5 \).
2. **\( f(x) = 6(x + 4)^2 + 10.5 \)**
This function opens upward (since the coefficient of the squared term is +6). Thus, the minimum value occurs at the vertex \( (−4, 10.5) \). This does not satisfy \( y \leq 10.5 \).
3. **\( f(x) = -6(x + 4)^2 + 10.5 \)**
This function opens downward (since the coefficient of the squared term is -6). The vertex is at \( (−4, 10.5) \), and the maximum value is 10.5, leading to the range \( y \leq 10.5 \). This satisfies the condition.
4. **\( f(x) = 10.5(x - 4)^2 - 6 \)**
This function opens upward (since the coefficient of the squared term is +10.5). Therefore, the minimum value occurs and this does not satisfy \( y \leq 10.5 \).
Thus, the quadratic function with the range \( y \leq 10.5 \) is:
**\( f(x) = -6(x + 4)^2 + 10.5 \)**