Answer:
To find the vertex, focus, directrix, and axis of symmetry of the given parabola \((x - 1)^2 = -3(y + 4)\), we first need to rewrite it in the standard form for a parabola that opens vertically.
The standard form for a parabola that opens either up or down is:
\[
(x - h)^2 = 4p(y - k)
\]
where \((h, k)\) is the vertex, \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix, but in the opposite direction).
Let's compare \((x - 1)^2 = -3(y + 4)\) to \((x - h)^2 = 4p(y - k)\):
1. **Vertex**: The vertex \((h, k)\) can be directly read from the equation. Here, \((h, k) = (1, -4)\).
2. **Value of \(p\)**: The coefficient of \((y - k)\) in the standard form is \(4p\). In the given equation, we have:
\[
(x - 1)^2 = -3(y + 4)
\]
This can be rewritten as:
\[
(x - 1)^2 = 4p(y - (-4))
\]
Comparing this with \((x - h)^2 = 4p(y - k)\), we see that \(4p = -3\). Therefore:
\[
p = \frac{-3}{4}
\]
3. **Focus**: The focus lies a distance \(p\) from the vertex along the axis of symmetry. Since \(p = -\frac{3}{4}\), this means the focus is \(\frac{3}{4}\) units below the vertex (since the parabola opens downward). Therefore, the focus is at:
\[
\left(1, -4 - \frac{3}{4}\right) = \left(1, -4.75\right)
\]
4. **Directrix**: The directrix is a line that is \(p\) units away from the vertex in the opposite direction of the focus. So, it will be \(\frac{3}{4}\) units above the vertex:
\[
y = -4 + \frac{3}{4} = -3.25
\]
Therefore, the directrix is the line:
\[
y = -3.25
\]
5. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex. Therefore, it is:
\[
x = 1
\]
In summary:
- **Vertex**: \((1, -4)\)
- **Focus**: \((1, -4.75)\)
- **Directrix**: \(y = -3.25\)
- **Axis of Symmetry**: \(x = 1\)
