To find which equation corresponds to a parabola with a vertex at [tex]\((-3, -2)\)[/tex], we need to understand the vertex form of a parabola. The vertex form is generally given as:
[tex]\[ x = a(y - k)^2 + h \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given that the vertex is [tex]\((-3, -2)\)[/tex], we need to determine which equation matches this vertex. Let's check each option:
### Option A: [tex]\( x = -2(y + 2)^2 - 3 \)[/tex]
Rewriting this in vertex form:
[tex]\[ x = -2(y - (-2))^2 - 3 \][/tex]
Comparing this with the standard vertex form [tex]\(x = a(y - k)^2 + h\)[/tex]:
- [tex]\(h = -3\)[/tex]
- [tex]\(k = -2\)[/tex]
This vertex [tex]\((-3, -2)\)[/tex] matches the given vertex.
### Option B: [tex]\( x = -2(y - 2)^2 - 3 \)[/tex]
Rewriting this in vertex form:
[tex]\[ x = -2(y - 2)^2 - 3 \][/tex]
Comparing this with the standard vertex form [tex]\(x = a(y - k)^2 + h\)[/tex]:
- [tex]\(h = -3\)[/tex]
- [tex]\(k = 2\)[/tex]
This vertex [tex]\((-3, 2)\)[/tex] does not match the given vertex.
### Option C: [tex]\( x = -2(y - 3)^2 - 2 \)[/tex]
Rewriting this in vertex form:
[tex]\[ x = -2(y - 3)^2 - 2 \][/tex]
Comparing this with the standard vertex form [tex]\(x = a(y - k)^2 + h\)[/tex]:
- [tex]\(h = -2\)[/tex]
- [tex]\(k = 3\)[/tex]
This vertex [tex]\((-2, 3)\)[/tex] does not match the given vertex.
### Option D: [tex]\( x = -2(y + 3)^2 - 2 \)[/tex]
Rewriting this in vertex form:
[tex]\[ x = -2(y - (-3))^2 - 2 \][/tex]
Comparing this with the standard vertex form [tex]\(x = a(y - k)^2 + h\)[/tex]:
- [tex]\(h = -2\)[/tex]
- [tex]\(k = -3\)[/tex]
This vertex [tex]\((-2, -3)\)[/tex] does not match the given vertex.
Among all the given options, only Option A [tex]\( x = -2(y + 2)^2 - 3 \)[/tex] has the vertex [tex]\((-3, -2)\)[/tex].
Thus, the correct equation is:
[tex]\[ \boxed{ x = -2(y + 2)^2 - 3 } \][/tex]
