Question

using the graph, determine the coordinates of the vertex of the parabola.

Explanation:

Step1: Recall vertex definition

The vertex of a parabola is its minimum (for upward - opening) or maximum (for downward - opening) point. For a parabola that opens upward (as this one does, since the arms go up), the vertex is the lowest point.

Step2: Find x - coordinate of vertex

The parabola is symmetric about the vertical line passing through its vertex. The roots of the parabola (where it crosses the x - axis) are at \(x = 4\) and \(x=8\). The x - coordinate of the vertex is the midpoint of the two roots. The formula for the midpoint of two numbers \(x_1\) and \(x_2\) is \(\frac{x_1 + x_2}{2}\). So, \(x=\frac{4 + 8}{2}=\frac{12}{2}=6\).

Step3: Find y - coordinate of vertex

Looking at the graph, the lowest point (vertex) has a y - coordinate of \(- 4\) (by observing the grid lines, the vertex is at \(y=-4\) when \(x = 6\)).

Answer:

The coordinates of the vertex of the parabola are \((6,-4)\)