Write an equation in intercept form of the parabola that passes through (-1,40) and x=-5, 4

At first I didn't understand what you meant by "x = -5, 4.  What I think you meant was "the horizontal intercepts of the graph of this parabola are (-5,0) and (4,0)."

If this is the case, then the equation of the parabola is found as follows:

y=ax^2 + bx + c   for the point (-5, 0) is 0=a(-5)^2 + b(-5) + c
                            for the point (4,0)   is  0 = a(4)^2 +b(4) + c
                            for the point (-1,40) is  40 = a(-1)^2 + b(-1) + c

Here we have 3 equations in 3 unknowns, which is enough to solve for {a,b,c}.  Using matrix algebra, I found that a= -2, b= -2, c= 40.

Then one equation for this parabola would be:

y = -2x^2 -  2x + 40

Check this by substitution.  Does the point (-5,0) satisfy y = -2x^2 -  2x + 40?
Yes.  So y = -2x^2 -  2x + 40  is the general form of the equation of this parabola.  To express it in intercept form, factor y = -2x^2 -  2x + 40.

Answers: the graph is translated down 1 unit and the domain and range of g(x) is the same as the domain and range of f(x).