Question

Derive the equation of a parabola given the focus is at (−4,6)
and the directrix is at y=8
. Fill in the missing values of the equation in standard form.(1 point)
y=−1/4x2− blank x+3

Answer 1

To derive the equation of the parabola with the given focus and directrix, we can follow these steps:

1. **Identify the properties of the parabola**:
- The focus is at \((-4, 6)\).
- The directrix is the line \(y = 8\).

2. **Find the vertex**:
- The vertex of a parabola is located halfway between the focus and the directrix. Since the focus is \((-4, 6)\) and the directrix is \(y=8\), we can find the vertex as follows:
- The y-coordinate of the vertex can be found by taking the average of the y-coordinate of the focus and the y-coordinate of the directrix:
\[
\text{Vertex y-coordinate} = \frac{6 + 8}{2} = 7
\]
- The x-coordinate of the vertex is the same as the x-coordinate of the focus since the directrix is horizontal:
\[
\text{Vertex} = (-4, 7)
\]

3. **Determine the direction of the parabola**:
- Since the focus is below the directrix, the parabola opens downwards.

4. **Determine the distance (p)**:
- \(p\) is the distance from the vertex to the focus (or from the vertex to the directrix). In this case:
\[
p = 7 - 6 = 1
\]

5. **Write the equation of the parabola**:
- For a parabola that opens downwards, the standard form of the equation is given by:
\[
(x - h)^2 = -4p(y - k)
\]
where \((h, k)\) is the vertex.

Substituting \(h = -4\), \(k = 7\), and \(p = 1\):
\[
(x + 4)^2 = -4(1)(y - 7)
\]
Simplifying:
\[
(x + 4)^2 = -4(y - 7)
\]

6. **Rewrite it in the form \(y = Ax^2 + Bx + C\)**:
Rearranging the equation:
\[
(x + 4)^2 = -4y + 28
\]
\[
-4y = (x + 4)^2 - 28
\]
\[
y = -\frac{1}{4}(x + 4)^2 + 7
\]
To convert this equation into the standard form \(y = Ax^2 + Bx + C\):
\[
y = -\frac{1}{4}(x^2 + 8x + 16) + 7
\]
\[
y = -\frac{1}{4}x^2 - 2x - 4 + 7
\]
\[
y = -\frac{1}{4}x^2 - 2x + 3
\]

7. **Fill in the missing values**:
The equation can be written as:
\[
y = -\frac{1}{4}x^2 - 2x + 3
\]
Thus, the value that needs to be filled in for \(- \text{blank}\) is \(2\):
\[
y = -\frac{1}{4}x^2 - 2x + 3
\]

The completed equation in standard form is:
\[
y = -\frac{1}{4}x^2 - 2x + 3
\]
Therefore, the missing value is **2**.