To solve this problem, let's understand the vertex form of a parabola. For a parabola that opens sideways (i.e. horizontal parabola), the vertex form is given by:
[tex]\[ x = a(y - k)^2 + h \][/tex]
Where [tex]\((h, k)\)[/tex] is the vertex.
Here, the vertex is given as [tex]\((-1, -3)\)[/tex]. We want to identify which equation among the given choices correctly reflects this vertex.
Let's examine each option:
### Option A: [tex]\( x = -2(y + 1)^2 - 3 \)[/tex]
- Rewrite it in the standard form:
[tex]\[ x = -2(y - (-1))^2 - 3 \][/tex]
- Here, [tex]\( h = -3 \)[/tex] and [tex]\( k = -1 \)[/tex].
The vertex [tex]\((-1, -3)\)[/tex] doesn't match when comparing [tex]\((h, k)\)[/tex].
### Option B: [tex]\( x = -2(y - 3)^2 - 1 \)[/tex]
- Rewrite it in the standard form:
[tex]\[ x = -2(y - 3)^2 - 1 \][/tex]
- Here, [tex]\( h = -1 \)[/tex] and [tex]\( k = 3 \)[/tex].
The vertex [tex]\((-1, 3)\)[/tex] doesn't match when comparing [tex]\((h, k)\)[/tex].
### Option C: [tex]\( x = -2(y - 1)^2 - 3 \)[/tex]
- Rewrite it in the standard form:
[tex]\[ x = -2(y - 1)^2 - 3 \][/tex]
- Here, [tex]\( h = -3 \)[/tex] and [tex]\( k = 1 \)[/tex].
The vertex [tex]\((-3, 1)\)[/tex] doesn't match when comparing [tex]\((h, k)\)[/tex].
### Option D: [tex]\( x = -2(y + 3)^2 - 1 \)[/tex]
- Rewrite it in the standard form:
[tex]\[ x = -2(y - (-3))^2 - 1 \][/tex]
- Here, [tex]\( h = -1 \)[/tex] and [tex]\( k = -3 \)[/tex].
The vertex [tex]\((-1, -3)\)[/tex] matches perfectly with the given vertex [tex]\((h, k)\)[/tex].
Given that the correct vertex [tex]\((-1, -3)\)[/tex] matches for option D only, the correct choice is:
[tex]\[ \boxed{D} \][/tex]
