To determine the equation of a parabola given its vertex, we need to understand the vertex form of a parabolic equation. The vertex form is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
For our case, the vertex is given as [tex]\((-5, 4)\)[/tex]. Substituting [tex]\(h = -5\)[/tex] and [tex]\(k = 4\)[/tex] into the vertex form, we get:
[tex]\[ y = a(x - (-5))^2 + 4 \][/tex]
[tex]\[ y = a(x + 5)^2 + 4 \][/tex]
Since the problem requires us to choose among options where the parabola opens downward, [tex]\(a\)[/tex] must be negative. Therefore, we have:
[tex]\[ y = -a(x + 5)^2 + 4 \][/tex]
Now, let's evaluate the given options:
Option A: [tex]\( y = -(x+5)^2 - 4 \)[/tex]
- The equation [tex]\( y = -(x+5)^2 - 4 \)[/tex] would represent a parabola with vertex at [tex]\((-5, -4)\)[/tex]. Hence, this is incorrect.
Option B: [tex]\( y = -(x-5)^2 + 4 \)[/tex]
- The equation [tex]\( y = -(x-5)^2 + 4 \)[/tex] would represent a parabola with vertex at [tex]\((5, 4)\)[/tex]. Hence, this is incorrect.
Option C: [tex]\( y = -(x-5)^2 - 4 \)[/tex]
- The equation [tex]\( y = -(x-5)^2 - 4 \)[/tex] would represent a parabola with vertex at [tex]\((5, -4)\)[/tex]. Hence, this is incorrect.
Option D: [tex]\( y = -(x+5)^2 + 4 \)[/tex]
- The equation [tex]\( y = -(x+5)^2 + 4 \)[/tex] would represent a parabola with vertex at [tex]\((-5, 4)\)[/tex]. This matches our derived equation.
Therefore, the correct equation for the parabola with vertex [tex]\((-5, 4)\)[/tex] is:
[tex]\[ \boxed{y = -(x+5)^2 + 4} \][/tex]
