Transformations of Functions

Mapping Rule

For a relation \(y = af(k(x - p)) + q\) there is a mapping rule of \((\frac{1}{k}x + p, ay + q)\)

Variable	Effect
a	Vertical compressions, stretches, and reflections
k	Horizontal compressions, stretches, and reflections
p	Horizontal translations (moves left and right)
q	Vertical translations (moves up and down)

Translations

Case	Effect
p > 0	Function is translated p units right
p < 0	Function is translated p units left
q > 0	Function is translated q units up
q < 0	Function is translated q units down

Stretches

Case	Effect
a > 1	Vertical expansion by a factor of a
0 < a < 1	Vertical compression by a factor of a
k > 1	Horizontal compression by a factor of \(\frac{1}{k}\)
0 < k < 1	Horizontal expansion by a factor of \(\frac{1}{k}\)

Reflections

Case	Effect
a > 0	No reflection
a < 0	Reflection about the x-axis
k > 0	No reflection
k < 0	Reflection about the y-axis

Side note: mathematically speaking, a and k is the same thing. The function can be algebraically manipulated to have a single a or a single k value to account for both. Communicationwise, they mean different things.

Describing

Start with the vertical and horizontal stretches. Describe the reflection as a separate entity from the stretches. Describe translations.