To identify which equation represents a parabola with the vertex at [tex]\((-5, 4)\)[/tex], we need to use the vertex form of a parabola equation, which is [tex]\(y = a(x - h)^2 + k\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given the vertex [tex]\((-5, 4)\)[/tex]:
- The value of [tex]\(h\)[/tex] is [tex]\(-5\)[/tex].
- The value of [tex]\(k\)[/tex] is [tex]\(4\)[/tex].
So the equation of the parabola should look like [tex]\(y = a(x + 5)^2 + 4\)[/tex].
Let's evaluate each option one by one to find out which matches the vertex form with the correct vertex location.
Option A: [tex]\(y = -(x + 5)^2 - 4\)[/tex]
- This equation is of the form [tex]\(y = a(x + 5)^2 - 4\)[/tex].
- The vertex of this equation would be [tex]\((-5, -4)\)[/tex], since [tex]\((x + 5)^2\)[/tex] shifts the vertex left by 5 and [tex]\(-4\)[/tex] shifts it down by 4.
- The vertex [tex]\((-5, -4)\)[/tex] does not match [tex]\((-5, 4)\)[/tex].
Option B: [tex]\(y = -(x - 5)^2 + 4\)[/tex]
- This equation is of the form [tex]\(y = a(x - 5)^2 + 4\)[/tex].
- The vertex of this equation would be [tex]\((5, 4)\)[/tex], since [tex]\((x - 5)^2\)[/tex] shifts the vertex right by 5 and [tex]\(+4\)[/tex] shifts it up by 4.
- The vertex [tex]\((5, 4)\)[/tex] does not match [tex]\((-5, 4)\)[/tex].
Option C: [tex]\(y = -(x - 5)^2 - 4\)[/tex]
- This equation is of the form [tex]\(y = a(x - 5)^2 - 4\)[/tex].
- The vertex of this equation would be [tex]\((5, -4)\)[/tex], since [tex]\((x - 5)^2\)[/tex] shifts the vertex right by 5 and [tex]\(-4\)[/tex] shifts it down by 4.
- The vertex [tex]\((5, -4)\)[/tex] does not match [tex]\((-5, 4)\)[/tex].
Option D: [tex]\(y = -(x + 5)^2 + 4\)[/tex]
- This equation is of the form [tex]\(y = a(x + 5)^2 + 4\)[/tex].
- The vertex of this equation would be [tex]\((-5, 4)\)[/tex], since [tex]\((x + 5)^2\)[/tex] shifts the vertex left by 5 and [tex]\(+4\)[/tex] shifts it up by 4.
- The vertex [tex]\((-5, 4)\)[/tex] matches the given vertex.
Therefore, the correct equation for the parabola with the vertex at [tex]\((-5, 4)\)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
