Final answer:
The vertex of the graph of the quadratic equation y=2(x-2)^2+4 is (2, 4). The form of the equation allows us to directly read the vertex as the equation is in vertex form, with a positive coefficient indicating that the parabola opens upwards.
Step-by-step explanation:
Finding the Vertex of a Quadratic Function
The student has asked about the vertex of the graph represented by the quadratic equation y = 2(x - 2)^2 + 4. This equation is in vertex form, which is generally written as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Here, a represents the coefficient that determines the width and direction of the parabola, while (h, k) represents the vertex of the parabola. In the given equation, the vertex can be directly read from the equation as the point (h, k).
To find the vertex, we observe that the equation is already in the completed square form, so the vertex (h, k) is the point where the parabola changes direction. Thus, the vertex of the given equation is at (2, 4) since h=2 and k=4. This means that the parabola opens upwards because the coefficient a=2 is positive, and the vertex (2, 4) is the lowest point on the graph.
Finding the vertex is essential in graphing parabolas as it provides a starting point and indicates the direction in which the parabola opens. In this case, since a is positive, the parabola opens upward, and the vertex represents the minimum point.