D.) reflect the graph about the x-axis and translate 4 units up
Explanation:
Before to identify the correct choice, let's see the definition of reflection along the x-axis and the y-axis:
1- A reflection of
about the x-axis can be done by changing
into
. This means that if we have a line in the form

a reflection about the x-axis can be done by changing the function into

2- A reflection of
about the y-axis can be done by changing
into
. This means that if we have a line in the form

a reflection about the x-axis can be done by replacing all the x with -x:

Back to our exercise:
The original function is
. The final function is
. First of all, we immediately notice that both the signs of m and q have been changed: therefore, it must be a reflection about the x-axis, so we can discard option B.
The reflection of f(x) about the x-axis is

We see that the y-intercept is +8, while in g(x) the y-intercept is +12. In order to match the two functions, we must translate f'(x) up by 4 units, so that we get

which corresponds to g(x). So, the correct option is D.