Making the y-values opposite in order to make the parabola concave down is the easy part. We simply have to multiply all of the y-values by -1.
The hard part is shifting the vertex so that it lies on the vertex of the original graph. We have options here. We can complete the square in order to find the vertex form of our equation. We can also take a shortcut to find the vertex of the equation. Let's take the shortcut. The x-value of the vertex of a parabola whose equation is can be found using the equation . In this case, we have . Now we can use this x value in our equation in order to find the y coordinate of the vertex. The work follows:
Notice the symmetry in our graphs above. The y-value of the vertex of our original purple graph is .
The y-value of the vertex of our flipped red graph is (You should verify this). In order to shift the red graph down so that its vertex conincides with the vertex of the purple graph, we will have to shift it down = = 10.25 units. Here is the equation and the graph of the solution.