Question

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

Explanation:

Step1: Recall vertex definition

The vertex of a parabola is the lowest (for upward - opening) or highest (for downward - opening) point. For a parabola, it's also on the axis of symmetry, which is the vertical line that splits the parabola into two mirror - image halves.

Step2: Analyze the graph

Looking at the given parabola (upward - opening), we can see that the vertex is the minimum point. From the graph, we can observe the coordinates of the vertex. First, find the x - coordinate: the axis of symmetry is the vertical line that passes through the mid - point of the two x - intercepts. The x - intercepts seem to be at \(x=-8\) and \(x = - 2\) (approximate from the graph). The mid - point formula for x - coordinate of the vertex (axis of symmetry) is \(x=\frac{x_1 + x_2}{2}=\frac{-8+( - 2)}{2}=\frac{-10}{2}=-5\). Then, looking at the y - coordinate of the vertex, from the graph, when \(x = - 5\), the y - coordinate of the minimum point (vertex) is \(-9\). So the vertex is at \((-5,-9)\).

Answer:

\((-5, - 9)\)