Without doing much work or manipulation of the function, we can use our knowledge of Vertex Form of quadratic functions, which is 

with  being the coordinates of the vertex. Knowing this, we can analyze our function  to find the vertex...     vertex: .

Note: This function is simply a transformation of the function

When transforming paraboloas, to translate up, add to the equation (or add to the Y).

To translate to the left, add to the X.

Don't forget that if you add to the X, then since X is squared, the addition to X must also be squared. 

 with the shift up 5 becomes: .

Now adding the shift to the left we get:

.

To move unit down, subtract from Y (or from the entire equation) , so subtract 1.

To move unit to the left, add to X (don't forget, that since you are squaring X, you must square the addition as well). 

With the move down our equation  becomes: .

Now to move it to the left we get .

The parent function for a parabolic function is  where  is the center of the parabola. To shift the parabola left of right, the value of h changes. Since there is a negative sign in the parent function, a positive value moves the parabola to the left and a negative value moves it to the right.  

Translations that effect x must be directly connected to x in the function and must also change the sign. So when the function was translated right two spaces, a  must be connected to the x value in the function. 

Translation that effect y must be directly connected to the constant in the funtion - so when the function was translated up 4 spaces a +4 must be added to the (-5) in the original function. 

When both of these happen in the function the new function must become: 

Because the parent function is , we can write the general form as:

.

a is the compression or stretch factor.

If , the function compresses or "narrows" by a factor of a.

If , the function stretches or "widens" by a factor of a.

b represents how the function shifts horizontally.

If b is negative, the function shifts to the left b units.

If b is positive, the function shifts to the right b units.

c represents how the function shifts vertically.

If c is positive, the function shifts up c units.

If c is negative, the function shifts down c units.

For our problem, a=3, b=-2, and c=5. (Remember that even though b is negative, the negative from the "general form" makes the sign positive). It follows that we have a compression by a factor of 3, a horizontal shift to the left 2 units, and a vertical shift up 5 units. 

To shift up 6 units, just add 6:

To shift to the right 3, subtract 3 from x:

First expand :

now this gives us:

distribute the 2 and the 4:

combine like terms:

Right now, the vertex for this point is , so to shift it up 4 and left 5 would place it at .

This gives us an equation of .

We could also get this by adding 5 inside the parentheses for the left 5 shift, and adding 4 on the outside for the up 4 shift.

Flipping this equation over the x-axis means that the sign of y changes.

The easiest way to accomplish this is just to multiply everything by -1.

To flip this over the -axis, the sign of x changes.

This entails changing to . 

.

Perhaps a simpler way to think about this is that the vertex for this parabola is at .

If we flip the equation over the y-axis, it will place the vertex at , making our new equation .