To determine the equation of a parabola that forms the arch in a bridge, given by the equation [tex]\( y = a(x-h)^2 + k \)[/tex], we need to identify the constants [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex]. The vertex form of a parabola provides valuable information about its shape and position.
Let's break down the components:
1. Vertex [tex]\((h, k)\)[/tex]: The vertex of the parabola is the highest point or lowest point of the arch, depending on the direction the parabola opens. In this case, since we're talking about an arch, you can assume the parabola opens downwards, and [tex]\( (h, k) \)[/tex] represents the highest point of the arch.
2. The constant [tex]\(a\)[/tex]: The value of [tex]\(a\)[/tex] determines how "wide" or "narrow" the parabola is. If [tex]\(a\)[/tex] is negative, the parabola opens downwards which is expected for an arch.
Without specific values for [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex], the generalized form of the equation remains as:
[tex]\[ y = a(x-h)^2 + k. \][/tex]
This form can describe the parabolic shape of the stone arch in a bridge. To get the exact equation, the specific values of [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] are necessary, which depend on the physical dimensions of the arch.
Therefore, the equation that describes the parabola formed by the arch is:
[tex]\[ y = a(x-h)^2 + k. \][/tex]
Here, [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] are specific constants that would need to be determined from additional information about the bridge's dimensions and the position of the arch.
