To determine which equation describes a parabola that opens left or right with its vertex at the point [tex]\((h, v)\)[/tex], we need to understand the general forms of parabolic equations.
1. Parabolas that open up or down:
The standard form for a parabola that opens up or down is:
[tex]\[
y = a(x - h)^2 + v
\][/tex]
Here, [tex]\((h, v)\)[/tex] is the vertex of the parabola.
2. Parabolas that open left or right:
The standard form for a parabola that opens left or right is:
[tex]\[
x = a(y - v)^2 + h
\][/tex]
Again, [tex]\((h, v)\)[/tex] is the vertex of the parabola.
Now, let's analyze the given options:
- Option A: [tex]\( y = a(x - v)^2 + h \)[/tex]
This describes a parabola that opens up or down, with its vertex at [tex]\((v, h)\)[/tex]. This equation does not fulfill the requirement of opening left or right. Therefore, it’s incorrect.
- Option B: [tex]\( x = a(v - h)^2 + v \)[/tex]
This form doesn’t correspond to any standard form of parabolic equations. Therefore, this option can be discarded as incorrect.
- Option C: [tex]\( y = a(x - h)^2 + v \)[/tex]
This also describes a parabola that opens up or down, with its vertex at [tex]\((h, v)\)[/tex]. This is not what we need since we are looking for a parabola that opens to the left or right.
- Option D: [tex]\( x = a(y - v)^2 + h \)[/tex]
This describes a parabola that opens to the left or right, with its vertex at [tex]\((h, v)\)[/tex]. This matches the requirement perfectly.
Thus, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
