To determine which equation represents a parabola with its vertex at [tex]\((5, -4)\)[/tex], we need to analyze the given options based on the standard form of a parabolic equation.
The vertex form of a parabolic equation is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given that the vertex is [tex]\((5, -4)\)[/tex], we can identify [tex]\(h = 5\)[/tex] and [tex]\(k = -4\)[/tex].
We need to check the vertex for each of the given equations:
Option A: [tex]\( y = 2(x + 5)^2 + 4 \)[/tex]
- The equation is in the form [tex]\( y = a(x - h)^2 + k \)[/tex].
- Here, [tex]\(h = -5\)[/tex] and [tex]\(k = 4\)[/tex].
- The vertex is [tex]\((-5, 4)\)[/tex], which does not match [tex]\((5, -4)\)[/tex].
Option B: [tex]\( y = 2(x - 5)^2 + 4 \)[/tex]
- The equation is in the form [tex]\( y = a(x - h)^2 + k \)[/tex].
- Here, [tex]\(h = 5\)[/tex] and [tex]\(k = 4\)[/tex].
- The vertex is [tex]\((5, 4)\)[/tex], which does not match [tex]\((5, -4)\)[/tex].
Option C: [tex]\( y = 2(x + 5)^2 - 4 \)[/tex]
- The equation is in the form [tex]\( y = a(x - h)^2 + k \)[/tex].
- Here, [tex]\(h = -5\)[/tex] and [tex]\(k = -4\)[/tex].
- The vertex is [tex]\((-5, -4)\)[/tex], which does not match [tex]\((5, -4)\)[/tex].
Option D: [tex]\( y = 2(x - 5)^2 - 4 \)[/tex]
- The equation is in the form [tex]\( y = a(x - h)^2 + k \)[/tex].
- Here, [tex]\(h = 5\)[/tex] and [tex]\(k = -4\)[/tex].
- The vertex is [tex]\((5, -4)\)[/tex], which matches the given vertex.
Thus, the correct equation for the parabola with vertex [tex]\((5, -4)\)[/tex] is:
[tex]\[ y = 2(x - 5)^2 - 4 \][/tex]
So the correct answer is:
[tex]\[ \boxed{D} \][/tex]
