Question

1. the vertex of a parabola is the point on a parabola that intersects the axis of symmetry. this is where the parabola changes direction. the vertex for ( d(x) ) is ( (4,2) ). the dolphin reaches a maximum height of 2 meters 4 seconds after the beachgoer begins watching. what is the vertex of ( b(x) ) and what does this point mean in context? is this a maximum or minimum?

Explanation:

Looking at the table, when \(x = 1\), \(y = 1.5\) (for \(B(x)\)). The vertex of a parabola is a point \((h,k)\) where the parabola changes direction. For \(B(x)\), from the table values, we can observe the pattern. As \(x\) increases from \(0\) to \(1\), \(y\) increases (from \(1\) to \(1.5\)), and as \(x\) increases from \(1\) to \(2\), \(y\) decreases (from \(1.5\) to \(1\)). So the vertex is \((1,1.5)\). In the context of the bird's path, the \(x\) - value represents time (in seconds) and the \(y\) - value represents height (in meters). Since the height increases before \(x = 1\) and decreases after \(x = 1\), this is a maximum point.

Answer:

The vertex of \(B(x)\) is \((1,1.5)\). This means that \(1\) second after the beachgoer begins watching, the bird reaches a maximum height of \(1.5\) meters. This is a maximum.