To find the standard form of the equation of a parabola that opens up or down, let's analyze the options presented.
### Step-by-Step Analysis
1. Consider the properties of a parabola that opens up or down:
- The general form for such a parabola is typically represented as [tex]\( y = a(x - h)^2 + k \)[/tex], where:
- [tex]\( (h, k) \)[/tex] is the vertex,
- [tex]\( a \)[/tex] determines the direction and the width of the parabola.
2. Evaluate Option A: [tex]\( y = a(x - h)^2 + v \)[/tex]
- This equation is in the form [tex]\( y = a(x - h)^2 + k \)[/tex] where [tex]\( k \)[/tex] is replaced by [tex]\( v \)[/tex] (both represent the vertical shift).
- This indeed represents the standard form of a parabola that opens up or down.
3. Evaluate Option B: [tex]\( y = ax^2 + bx + c \)[/tex]
- This equation is in the general quadratic form in [tex]\( y \)[/tex] and [tex]\( x \)[/tex], also known as the vertex form when rearranged and completed the square.
- However, it is not explicitly structured to highlight the vertex. It might represent a parabola, but we're specifically looking for the "standard form" we defined.
4. Evaluate Option C: [tex]\( x = a(y - v)^2 + h \)[/tex]
- This equation mirrors the typical form of a parabola that opens left or right.
- This is not the standard form for a parabola that opens upward or downward.
5. Evaluate Option D: [tex]\( x = ay^2 + by + c \)[/tex]
- Similar to the reasoning for Option C, this form is more aligned with a parabolic curve opening toward the left or right.
- Therefore, it’s not the standard form for a parabola that opens upward or downward.
### Conclusion
After analyzing each option, the correct answer that matches the standard form of the equation for a parabola opening up or down, which is [tex]\( y = a(x - h)^2 + k \)[/tex], aligns with:
Option A: [tex]\( y = a(x - h)^2 + v \)[/tex]
Thus, the standard form of the equation of a parabola that opens up or down is given by Option A.
