Transformations of Functions
Given the graph of f(x) and a constants h, k, A, and B the graph of g(x) can be created by transforming
the graph of f(x) as follows:
g(x) = f(x – h)
a horizontal shift right h units
g(x) = f(x + h)
a horizontal shift left h units
g(x) = f(x) + k
a vertical shift up k units
g(x) = f(x) – k
a vertical shift down k units
g(x) = f(-x)
a reflection about the y-axis
g(x) = -f(x)
a reflection about the x-axis
a vertical stretch or compression
 stretched by a factor of A,
if A > 1
 compressed by a factor of A,
if 0 < A < 1
 stretched/compressed by a factor of |A| and reflected across xaxis, if A < 0
g(x) = Af(x)
a horizontal stretch or compression
 compressed by a factor of 1/B,
if B > 1
 stretched by a factor of 1/B,
if 0 < B < 1
 stretched/compressed by a factor of |1/B| and reflected across
y-axis, if B < 0
g(x) = f(Bx)
Combinations of Transformations
For nonzero constants, A, B, h and k, the graph of the function
y = Af(B(x – h)) + k
is obtained by applying the transformations to the graphs of f(x) in the following order:
First:
horizontal stretch/compression by a factor of 1/|B|
Second:
horizontal shift by h units
Third:
vertical stretch/compression by a factor of |A|
Fourth:
vertical shift by k units

If A < 0, follow the vertical stretch/compression by a reflection about the x-axis.

If B < 0, follow the horizontal stretch/compression by a reflection about the y-axis.