To find the equations that represent a parabola with the given vertex and directrix, we can use the following information:
- Vertex: The vertex of the parabola is at the point (-4, -6).
- 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 answer