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
?(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
Answers
Answered by
GPT-4o mini
To determine which quadratic function has a range of \( y \leq 10.5 \), we need to examine the leading coefficient and the vertex of each function.
The general form of a quadratic function is given by:
\[
f(x) = a(x-h)^2 + k
\]
Here, the vertex of the parabola is at the point \( (h, k) \). The leading coefficient \( a \) determines the direction of the parabola:
- If \( a > 0 \), the parabola opens upwards, and the range will be \( [k, \infty) \).
- If \( a < 0 \), the parabola opens downwards, and the range will be \( (-\infty, k] \).
Now, let's evaluate the given options:
1. **\( f(x) = -6(x + 4)^2 + 10.5 \)**
- Here, \( a = -6 \) (which is negative), and the vertex is at \( ( -4, 10.5 ) \). The range is \( y \leq 10.5 \). This function fits our requirement.
2. **\( f(x) = 10.5(x - 4)^2 - 6 \)**
- Here, \( a = 10.5 \) (which is positive), so the range is \( y \geq -6 \). This does not work.
3. **\( f(x) = -6(x - 10.5)^2 + 4 \)**
- Here, \( a = -6 \) (which is negative), and the vertex is at \( (10.5, 4) \). The range is \( y \leq 4 \). This does not meet the requirement.
4. **\( f(x) = 6(x + 4)^2 + 10.5 \)**
- Here, \( a = 6 \) (which is positive), so the range is \( y \geq 10.5 \). This does not work.
The only function from the options that satisfies the condition \( y \leq 10.5 \) is:
**\( f(x) = -6(x + 4)^2 + 10.5 \)**.
The general form of a quadratic function is given by:
\[
f(x) = a(x-h)^2 + k
\]
Here, the vertex of the parabola is at the point \( (h, k) \). The leading coefficient \( a \) determines the direction of the parabola:
- If \( a > 0 \), the parabola opens upwards, and the range will be \( [k, \infty) \).
- If \( a < 0 \), the parabola opens downwards, and the range will be \( (-\infty, k] \).
Now, let's evaluate the given options:
1. **\( f(x) = -6(x + 4)^2 + 10.5 \)**
- Here, \( a = -6 \) (which is negative), and the vertex is at \( ( -4, 10.5 ) \). The range is \( y \leq 10.5 \). This function fits our requirement.
2. **\( f(x) = 10.5(x - 4)^2 - 6 \)**
- Here, \( a = 10.5 \) (which is positive), so the range is \( y \geq -6 \). This does not work.
3. **\( f(x) = -6(x - 10.5)^2 + 4 \)**
- Here, \( a = -6 \) (which is negative), and the vertex is at \( (10.5, 4) \). The range is \( y \leq 4 \). This does not meet the requirement.
4. **\( f(x) = 6(x + 4)^2 + 10.5 \)**
- Here, \( a = 6 \) (which is positive), so the range is \( y \geq 10.5 \). This does not work.
The only function from the options that satisfies the condition \( y \leq 10.5 \) is:
**\( f(x) = -6(x + 4)^2 + 10.5 \)**.