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 break down the standard form of such a parabola.
A parabola that opens up or down has an equation in the form:
[tex]\[ y = a(x - h)^2 + v \][/tex]
Here's a detailed explanation:
1. Vertex Form of a Parabola Opening Up or Down: The general form of a parabola that opens either upwards or downwards is given by [tex]\( y = a(x - h)^2 + v \)[/tex], where:
- [tex]\((h, v)\)[/tex] is the vertex of the parabola.
- [tex]\(a\)[/tex] is a constant that affects the width and direction of the parabola. If [tex]\(a > 0\)[/tex], the parabola opens upwards, and if [tex]\(a < 0\)[/tex], the parabola opens downwards.
2. Understanding Each Term:
- [tex]\( h \)[/tex]: The [tex]\(x\)[/tex]-coordinate of the vertex.
- [tex]\( v \)[/tex]: The [tex]\(y\)[/tex]-coordinate of the vertex.
- [tex]\( (x - h) \)[/tex]: This term indicates that the parabola is shifted horizontally to the point [tex]\(h\)[/tex].
- [tex]\( a(x - h)^2 \)[/tex]: This term represents the quadratic nature of the equation, ensuring it's a parabola.
3. Comparing Options:
- Option A: [tex]\( x = a(y - h)^2 + v \)[/tex] - This describes a parabola opening to the left or right, not up or down.
- Option B: [tex]\( y = a(x - v)^2 + h \)[/tex] - This incorrectly places the vertex's coordinates as [tex]\((v, h)\)[/tex] and is not in the standard form.
- Option C: [tex]\( y = a(x - h)^2 + v \)[/tex] - This matches our standard form.
- Option D: [tex]\( x = a(y - v)^2 + h \)[/tex] - This describes a parabola opening to the left or right, not up or down.
Based on the standard form we identified, the correct equation that describes a parabola opening up or down with the vertex at [tex]\((h, v)\)[/tex] is:
[tex]\[ y = a(x - h)^2 + v \][/tex]
Thus, the correct answer is:
C. [tex]\( y = a(x - h)^2 + v \)[/tex]
