Question

the graph represents the potential area of a concrete rectangle, based on length and width. which inequality in vertex form represents the graphed region? ( y < - 2 ( x - 16 ) ^ { 2 } + 32 ) ( y < - \frac { 1 } { 2 } ( x - 16 ) ^ { 2 } + 32 ) ( y < - 2 ( x + 16 ) ^ { 2 } - 32 ) ( y < - \frac { 1 } { 2 } ( x + 16 ) ^ { 2 } - 32 )

Explanation:

Step1: Recall the vertex form of a parabola

The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h,k)\) is the vertex of the parabola.
From the graph, the vertex is \((16,32)\), so \(h = 16\) and \(k = 32\).
The inequality will be of the form \(y< a(x - 16)^2+32\) (since the region is below the parabola).

Step2: Find the value of \(a\)

Substitute the point \((12,24)\) into the equation \(y=a(x - 16)^2+32\).
\(24=a(12 - 16)^2+32\)
\(24=a(-4)^2+32\)
\(24 = 16a+32\)
\(16a=24 - 32\)
\(16a=-8\)
\(a=-\frac{8}{16}=-\frac{1}{2}\)

Answer:

\(y<-\frac{1}{2}(x - 16)^2+32\) (the second option)