How to Combine Various Transformations
Certain mathematical expressions allow you to combine stretching, shrinking, translating, and reflecting a function all into one graph. An expression that shows all the transformations in one is
where

a is the vertical transformation.

c is the horizontal transformation.

h is the horizontal shift.

v is the vertical shift.
For instance, f(x) = –2(x – 1)^{2} + 4 moves the graph of y = x^{2} right 1 unit, vertically stretches it by a factor of 2, reflects it upside down, and then moves it up 4 units.
This figure shows each stage.

Figure a is the parent graph: k(x) = x^{2}.

Figure b is the horizontal shift to the right by one: h(x) = (x – 1)^{2}.

Figure c is the vertical stretch of two: f(x) = –2(x – 1)^{2}. (Notice that because the value was negative, the graph was also turned upside down.)

Figure d is the vertical shift up by four: g(x) = –2(x – 1)^{2} + 4.
The following transformation illustrates the importance of the order of the process. You graph the function
with the following steps:

Rewrite the function in the form
Reorder the function so that the x comes first (in descending order). And don’t forget the negative sign! Here it is:

Factor out the coefficient in front of the x.
You now have

Reflect the parent graph.
Because the –1 is inside the squareroot function, q(x) is a horizontal reflection over a vertical line of

Shift the graph.
The factored form of q(x) (from Step 2) reveals that the horizontal shift is four to the right.
This figure shows the graph of