To determine which equations below could represent a parabola with its vertex at the point [tex]\((5, -3)\)[/tex], we need to analyze each of the given options in detail.
### Step-by-Step Analysis
Key Information:
- The given vertex for the parabola is [tex]\((5, -3)\)[/tex].
- The forms of equations of parabolas are:
- [tex]\(y = a(x-h)^2 + k\)[/tex] for parabolas that open upward or downward (where the vertex is [tex]\((h, k)\)[/tex]).
- [tex]\(x = a(y-k)^2 + h\)[/tex] for parabolas that open leftward or rightward (where the vertex is [tex]\((h, k)\)[/tex]).
### Option A: [tex]\(y = -3(x-5)^2 - 3\)[/tex]
This equation is in the form [tex]\(y = a(x-h)^2 + k\)[/tex].
- Here, [tex]\(h = 5\)[/tex] and [tex]\(k = -3\)[/tex].
- Thus, the vertex is [tex]\((5, -3)\)[/tex].
Conclusion: This equation could be the one for the parabola.
### Option B: [tex]\(x = 3(y+3)^2 + 5\)[/tex]
This equation is in the form [tex]\(x = a(y-k)^2 + h\)[/tex].
- Here, [tex]\(k = -3\)[/tex] and [tex]\(h = 5\)[/tex].
- Thus, the vertex is [tex]\((5, -3)\)[/tex].
Conclusion: This equation could be the one for the parabola.
### Option C: [tex]\(x = -3(y+3)^2 + 5\)[/tex]
This equation is also in the form [tex]\(x = a(y-k)^2 + h\)[/tex].
- Here, [tex]\(k = -3\)[/tex] and [tex]\(h = 5\)[/tex].
- Thus, the vertex is [tex]\((5, -3)\)[/tex].
Conclusion: This equation could be the one for the parabola.
### Option D: [tex]\(x = 3(y-5)^2 - 3\)[/tex]
This equation is in the form [tex]\(x = a(y-k)^2 + h\)[/tex].
- Here, [tex]\(k = 5\)[/tex] and [tex]\(h = -3\)[/tex].
- Thus, the vertex is [tex]\((-3, 5)\)[/tex].
Conclusion: This equation does not match the given vertex and is, therefore, not a possible equation for the parabola.
### Final Answer:
The equations that could represent the given parabola with vertex [tex]\((5, -3)\)[/tex] are:
- [tex]\(A. \ y = -3(x-5)^2 - 3\)[/tex]
- [tex]\(B. \ x = 3(y+3)^2 + 5\)[/tex]
- [tex]\(C. \ x = -3(y+3)^2 + 5\)[/tex]
Therefore, the possible options are:
[tex]\[ \boxed{[1, 2, 3]} \][/tex]
