To answer this question, we need to remember that:
1. If a function has been reflected in the x-axis, we have:
[tex]-f(x)[/tex]2. If a function has been translated by h units to the right, we have:
[tex]f(x-h)[/tex]Therefore, if we have these two transformations on the function:
[tex]f(x)=-2x^2+6[/tex]Then, if we apply the two transformations above, we have:
1. Reflection over the x-axis:
[tex]F(x)=-2x^2+6\Rightarrow-F(x)=-(-2x^2+6)[/tex]Therefore
[tex]-F(x)=2x^2-6[/tex]2. If we translate the function h units to the right, then we have:
[tex]-F(x-h)=2(x-h)^2-6[/tex]If we observe the options, we have that if we translate the function 5 units to the right, we have:
[tex]-F(x-5)=2(x-5)^2-6[/tex]In summary, therefore, we have that the function which describes the resulting transformation is:
[tex]H(x)=2(x-5)^2-6[/tex][Option C.]
We can see this in the following graph: the red parabola is the original function. The green parabola is the result of the transformation:
