Start by visualizing the graph associated with the function :

Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of  looks like this:

Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function  :

The  portion results in the graph being shifted 3 units to the left, while the  results in the graph being shifted six units down.  Vertical shifts are the same sign as the number outside the parentheses, while horizontal shifts are the OPPOSITE direction as the sign inside the parentheses, associated with .

The function  shifts a function f(x) units to the left. Conversely,  shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function  or .

The parent function of a parabola is where are the vertex.

The original graph of a parabolic (quadratic) function has a vertex at (0,0) and shifts left or right by h units and up or down by k units.

.

This function then shifts 1 unit left, and 4 units down, and the negative in front of the squared term denotes a rotation over the x-axis.

Correct Answer:

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.