To determine which equation represents a parabola with its vertex at [tex]\((-5, 4)\)[/tex], we need to consider the vertex form of a quadratic equation. The vertex form is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex. For our problem, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-5, 4)\)[/tex]. Therefore:
- [tex]\( h = -5 \)[/tex]
- [tex]\( k = 4 \)[/tex]
Substitute the vertex coordinates into the vertex form:
[tex]\[ y = a(x + 5)^2 + 4 \][/tex]
Here, [tex]\( (x - (-5)) \)[/tex] simplifies to [tex]\( (x + 5) \)[/tex]. The coefficient [tex]\( a \)[/tex] determines the direction and width of the parabola. Since the problem indicates that the parabola opens downwards, [tex]\( a \)[/tex] must be negative.
We now need to match this equation to one of the given options:
A. [tex]\( y = -(x - 5)^2 + 4 \)[/tex]
B. [tex]\( y = -(x - 5)^2 - 4 \)[/tex]
C. [tex]\( y = -(x + 5)^2 + 4 \)[/tex]
D. [tex]\( y = -(x + 5)^2 - 4 \)[/tex]
Among these options, option C reflects [tex]\( a < 0 \)[/tex], [tex]\( h = -5 \)[/tex], and [tex]\( k = 4 \)[/tex] correctly:
[tex]\[ y = -(x + 5)^2 + 4 \][/tex]
Thus, the equation that matches a parabola with vertex [tex]\((-5, 4)\)[/tex] is:
Answer: C. [tex]\( y = -(x + 5)^2 + 4 \)[/tex]
