To write the quadratic function of the parabola given its vertex [tex]\((-3, 1)\)[/tex] and directrix [tex]\(y = 6\)[/tex], we follow these steps:
1. Identify the distance [tex]\(p\)[/tex]:
- The directrix is at [tex]\(y=6\)[/tex] and the vertex is at [tex]\((-3, 1)\)[/tex].
- The distance [tex]\(p\)[/tex] between the vertex and the directrix is given by:
[tex]\[
p = \frac{6 - 1}{2} = \frac{5}{2} = 2.5
\][/tex]
2. Determine the focus:
- Since the directrix is above the vertex, the focus is [tex]\(p\)[/tex] units below the vertex.
- The vertex is at [tex]\(y = 1\)[/tex], so the focus is at [tex]\(1 - 2.5 = -1.5\)[/tex].
3. Set up the standard form equation of the parabola:
- The standard form of a parabola that opens downward (since the focus is below the vertex and p-value is negative) is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
- Here, [tex]\((h, k)\)[/tex] is the vertex [tex]\((-3, 1)\)[/tex] and [tex]\(p = -2.5\)[/tex].
4. Substitute the values into the equation:
- Given [tex]\(h = -3\)[/tex], [tex]\(k = 1\)[/tex], and [tex]\(p = -2.5\)[/tex]:
[tex]\[
(x + 3)^2 = 4(-2.5)(y - 1)
\][/tex]
- Simplifying this, we get:
[tex]\[
(x + 3)^2 = -10(y - 1)
\][/tex]
Thus, the equation of the parabola is:
[tex]\[
f(x) = (x + 3)^2 = -10(y - 1)
\][/tex]
