Question

Olivia is asked to graph a parabola on a coordinate plane. The focus is located at (0, -3), and the directrix is at y = 1. Fill in the missing values on the table for the coordinates of the parabola. (1 point)

x-value y-value

2

0

-4

-1

74

-4

Answer 1

To find the equation of the parabola given the focus at \((0, -3)\) and the directrix at \(y = 1\), we can determine its orientation and vertex.

1. **Determine the vertex**: The vertex of the parabola is located halfway between the focus and directrix. The focus is at \((0, -3)\) and the directrix is a horizontal line at \(y = 1\).

The y-coordinate of the vertex can be found by averaging the focus and the directrix:
\[
\text{Vertex } y = \frac{-3 + 1}{2} = \frac{-2}{2} = -1
\]
The x-coordinate of the vertex is the same as that of the focus (since the focus and directrix are both vertical):
\[
\text{Vertex } = (0, -1)
\]

2. **Determine the equation of the parabola**: Since the parabola opens downward (the focus is below the directrix), we can use the standard form for a vertical parabola:
\[
(x - h)^2 = 4p(y - k)
\]
where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.

From the vertex \((0, -1)\) to the focus \((0, -3)\), we see:
\[
p = -3 - (-1) = -2 \quad (\text{since } p \text{ is negative for downward opening})
\]

So, the equation becomes:
\[
(x - 0)^2 = 4(-2)(y - (-1))
\]
Simplifying this gives:
\[
x^2 = -8(y + 1)
\]

Rearranging:
\[
y = -\frac{1}{8}x^2 - 1
\]

3. **Populate the table values**:
Now, we can plug the x-values into the equation to find the y-values.

- For \(x = 2\):
\[
y = -\frac{1}{8}(2^2) - 1 = -\frac{1}{8}(4) - 1 = -\frac{4}{8} - 1 = -\frac{1}{2} - 1 = -\frac{3}{2}
\]

- For \(x = 0\):
\[
y = -\frac{1}{8}(0^2) - 1 = -1
\]

- For \(x = -4\):
\[
y = -\frac{1}{8}(-4^2) - 1 = -\frac{1}{8}(16) - 1 = -2 - 1 = -3
\]

- For \(x = -1\):
\[
y = -\frac{1}{8}(-1^2) - 1 = -\frac{1}{8}(1) - 1 = -\frac{1}{8} - 1 = -\frac{1}{8} - \frac{8}{8} = -\frac{9}{8}
\]

- For \(x = 74\):
\[
y = -\frac{1}{8}(74^2) - 1 = -\frac{1}{8}(5476) - 1 = -684.5 - 1 = -685.5
\]

- For \(x = -4\) (again):
The corresponding value is already calculated as \(y = -3\).


Now the table would look like this:

\[
\begin{array}{|c|c|}
\hline
\text{x-value} & \text{y-value} \\
\hline
2 & -\frac{3}{2} \\
0 & -1 \\
-4 & -3 \\
-1 & -\frac{9}{8} \\
74 & -685.5 \\
-4 & -3 \\
\hline
\end{array}
\]

Feel free to ask if you need any clarification or further assistance!