Question

The vertex (V) and the directrix (d) of a parabola are shown.



The vertex is at (negative 1, 2). The directrix is the line y equals 0.

Question
Choose ,begin emphasis,all,end emphasis, of the equations which represent the parabola.

Answer options with 6 options
1.
4 y equals x squared plus 2 x plus 5

2.
8 y equals x squared minus 4 x minus 4

3.
8 y equals x squared plus 2 x plus 17

4.
left-parenthesis x plus 1 right-parenthesis squared equals 4 left-parenthesis y minus 1 right-parenthesis

Answer 1

To find the equations that represent the parabola given the vertex and directrix, we can use the properties of parabolas.

Given the vertex at \((-1, 2)\) and the directrix \(y = 0\), we can determine the orientation and the equation of the parabola.

1. **Orientation**: Since the vertex is above the directrix (the vertex \(y\) value is 2, while the directrix \(y\) value is 0), the parabola opens upwards.

2. **Standard Form**: The standard form of a parabola that opens upwards is given by:

\[
(x - h)^2 = 4p(y - k)
\]

where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the directrix. In this case:

- \((h, k) = (-1, 2)\)
- The distance \(p\) is \(2\) (from \(2\) to \(0\)).

Thus, \(p = 2\), and \(4p = 8\).

Substituting in the values, we have:

\[
(x + 1)^2 = 8(y - 2)
\]

Rearranging gives us:

\[
(x + 1)^2 = 8y - 16 \quad \text{or} \quad (x + 1)^2 - 8y + 16 = 0
\]

This shows us the standard form of our parabola with the vertex at \((-1, 2)\) and a directrix at \(y = 0\).

Now, we analyze the answer options provided:

1. **Option 1**: \(4y = x^2 + 2x + 5\)
- Rearranging gives \(y = \frac{1}{4}(x^2 + 2x + 5)\). This does not fit our vertex or directrix.

2. **Option 2**: \(8y = x^2 - 4x - 4\)
- Rearranging gives \(y = \frac{1}{8}(x^2 - 4x - 4)\). This also does not fit our vertex or directrix.

3. **Option 3**: \(8y = x^2 + 2x + 17\)
- Rearranging gives \(y = \frac{1}{8}(x^2 + 2x + 17)\). This also does not fit our vertex or directrix.

4. **Option 4**: \((x + 1)^2 = 4(y - 2)\)
- Rearranging gives \((x + 1)^2 = 8y - 16\), which confirms that this option is exactly of the expected form of the parabola.

Therefore, the only option that represents the parabola is:

**Option 4**: \((x + 1)^2 = 4(y - 2)\).

Thus, the only correct answer is option 4.