Question
Choose ,begin emphasis,all,end emphasis, of the equations which represent the parabola with vertex the point negative 4 comma negative 6 and directrix y equals negative 2.
Answer options with 6 options
A.
left-parenthesis x plus 4 right-parenthesis squared equals negative 8 left-parenthesis y plus 4 right-parenthesis
B.
left-parenthesis x plus 4 right-parenthesis squared equals negative 16 left-parenthesis y plus 6 right-parenthesis
C.
left-parenthesis x plus 6 right-parenthesis squared equals negative 16 left-parenthesis y plus 4 right-parenthesis
D.
8 y equals negative x squared minus 8 x minus 48
E.
16 y equals negative x squared minus 12 x minus 100
F.
16 y equals negative x squared minus 8 x minus 112
Answer options with 6 options
A.
left-parenthesis x plus 4 right-parenthesis squared equals negative 8 left-parenthesis y plus 4 right-parenthesis
B.
left-parenthesis x plus 4 right-parenthesis squared equals negative 16 left-parenthesis y plus 6 right-parenthesis
C.
left-parenthesis x plus 6 right-parenthesis squared equals negative 16 left-parenthesis y plus 4 right-parenthesis
D.
8 y equals negative x squared minus 8 x minus 48
E.
16 y equals negative x squared minus 12 x minus 100
F.
16 y equals negative x squared minus 8 x minus 112
Answers
GPT-4o mini
To find the equations that represent a parabola with the given vertex and directrix, we can use the following information:
1. **Vertex:** The vertex of the parabola is at the point (-4, -6).
2. **Directrix:** The equation of the directrix is \( y = -2 \).
### Properties of the Parabola:
A parabola opens upward or downward depending on the position of the vertex and the directrix. Since the vertex (-4, -6) is below the directrix (-2), the parabola opens downwards.
### Finding the Equation:
The standard form of a parabola that opens downwards is given by the equation:
\[
(x - h)^2 = -4p(y - k)
\]
where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the directrix.
### Calculate \(p\):
Distance \(p\) from the vertex (-4, -6) to the directrix \(y = -2\):
\[
p = |-2 - (-6)| = |-2 + 6| = |4| = 4
\]
Since the parabola opens downwards, \(p\) will be negative in our formula, hence \(4p = -16\).
### Substituting Values:
Now substituting \(h = -4\) and \(k = -6\) into the equation:
\[
(x + 4)^2 = -16(y + 6)
\]
This gives us the equation in standard form:
\[
(x + 4)^2 = -16(y + 6)
\]
### Examining the Answer Options:
Looking for answers that match \( (x + 4)^2 = -16(y + 6) \):
- **A.** \( (x + 4)^2 = -8(y + 4) \) (Not correct)
- **B.** \( (x + 4)^2 = -16(y + 6) \) (Correct)
- **C.** \( (x + 6)^2 = -16(y + 4) \) (Not correct)
- **D.** \( 8y = -x^2 - 8x - 48 \) (Not correct)
- **E.** \( 16y = -x^2 - 12x - 100 \) (Not correct)
- **F.** \( 16y = -x^2 - 8x - 112 \) (Not correct)
From the evaluation, the only correct option that represents the parabola with the given vertex and directrix is:
**B.** \( (x + 4)^2 = -16(y + 6) \)
1. **Vertex:** The vertex of the parabola is at the point (-4, -6).
2. **Directrix:** The equation of the directrix is \( y = -2 \).
### Properties of the Parabola:
A parabola opens upward or downward depending on the position of the vertex and the directrix. Since the vertex (-4, -6) is below the directrix (-2), the parabola opens downwards.
### Finding the Equation:
The standard form of a parabola that opens downwards is given by the equation:
\[
(x - h)^2 = -4p(y - k)
\]
where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the directrix.
### Calculate \(p\):
Distance \(p\) from the vertex (-4, -6) to the directrix \(y = -2\):
\[
p = |-2 - (-6)| = |-2 + 6| = |4| = 4
\]
Since the parabola opens downwards, \(p\) will be negative in our formula, hence \(4p = -16\).
### Substituting Values:
Now substituting \(h = -4\) and \(k = -6\) into the equation:
\[
(x + 4)^2 = -16(y + 6)
\]
This gives us the equation in standard form:
\[
(x + 4)^2 = -16(y + 6)
\]
### Examining the Answer Options:
Looking for answers that match \( (x + 4)^2 = -16(y + 6) \):
- **A.** \( (x + 4)^2 = -8(y + 4) \) (Not correct)
- **B.** \( (x + 4)^2 = -16(y + 6) \) (Correct)
- **C.** \( (x + 6)^2 = -16(y + 4) \) (Not correct)
- **D.** \( 8y = -x^2 - 8x - 48 \) (Not correct)
- **E.** \( 16y = -x^2 - 12x - 100 \) (Not correct)
- **F.** \( 16y = -x^2 - 8x - 112 \) (Not correct)
From the evaluation, the only correct option that represents the parabola with the given vertex and directrix is:
**B.** \( (x + 4)^2 = -16(y + 6) \)