To determine the equation that describes a parabola that opens left or right and whose vertex is at the point [tex]\((h, v)\)[/tex], let’s analyze each option provided.
First, recall the general forms of parabolas:
- For a parabola that opens up or down, the general form is:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
where [tex]\((h, k)\)[/tex] represents the vertex.
- For a parabola that opens left or right, the general form is:
[tex]\[
x = a(y - k)^2 + h
\][/tex]
where [tex]\((h, k)\)[/tex] represents the vertex.
Given this, the parabola that opens left or right fits the second form. The vertex given in the question is [tex]\((h, v)\)[/tex]. Thus, substituting [tex]\(h\)[/tex] and [tex]\(v\)[/tex] into the standard form [tex]\(x = a(y - k)^2 + h\)[/tex], we need to check which choice matches this form.
Let’s go through each option:
A. [tex]\(x = a(y - h)^2 + v\)[/tex]
- Here, [tex]\(h\)[/tex] is subtracted from [tex]\(y\)[/tex], making the vertex component incorrect compared to the standard form [tex]\(x = a(y - v)^2 + h\)[/tex].
B. [tex]\(x = a(y - y)^2 + h\)[/tex]
- This expression simplifies to [tex]\(x = h\)[/tex], which does not represent a parabola at all.
C. [tex]\(y = a(x - h)^2 + h\)[/tex]
- This equation is in the form of a parabola that opens up or down, which does not match the requirement.
D. [tex]\(y = a(x - h)^2 + v\)[/tex]
- This equation also describes a parabola that opens up or down, which is incorrect for our conditions.
Thus, the correct answer must be:
A. [tex]\(x = a(y - h)^2 + v\)[/tex]
