To determine which equation describes a parabola that opens up or down and whose vertex is at the point [tex]\((h, v)\)[/tex], let's recall some important concepts about parabolas.
The standard form of a parabolic equation that opens up or down is given as:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, the vertex of the parabola is at the point [tex]\((h, k)\)[/tex]. The coefficient [tex]\(a\)[/tex] determines the direction and the width of the parabola:
- If [tex]\(a > 0\)[/tex], the parabola opens upwards.
- If [tex]\(a < 0\)[/tex], the parabola opens downwards.
Given this form, we need to match it to one of the provided options, with the vertex [tex]\((h, v)\)[/tex]:
A. [tex]\(x = a(y - v)^2 + h\)[/tex] – This form does not represent a parabola that opens up or down; it represents a parabola that opens to the left or right.
B. [tex]\(x = a(y - h)^2 + v\)[/tex] – This is also a form that represents a parabola that opens to the left or right.
C. [tex]\(y = a(x - v)^2 + h\)[/tex] – This form indicates a parabola with a vertex at [tex]\((v, h)\)[/tex], but not in the correct orientation relative to the vertex provided.
D. [tex]\(y = a(x - h)^2 + v\)[/tex] – This correctly represents a parabola that opens up or down with the vertex at [tex]\((h, v)\)[/tex].
Thus, the correct equation that describes a parabola opening up or down with the vertex at the point [tex]\((h, v)\)[/tex] is:
[tex]\[ \boxed{D. \, y = a(x - h)^2 + v} \][/tex]
