Transformations of Maps and Functions
There is a beautiful and simple way to understand functions in general. In some sense, this is the lion’s share of what you are expected to take away from an algebra I and II course. You look at function and you “read” it. That is, you visualize the graph and relevant details to a greater or lesser extent (depending on your level of reading comprehension)!
1. General Transformations of a Function or Map
We will consider some parent function (or map) $f(x)$. The series of transformation that we apply to $f(x)$ can be written as
$$af(b(x-h)) + k$$
a is the vertical dilation, b is the horizontal dilation, h is the horizontal translation (to the right), k is the vertical translation (upwards). The mathematical term dilation refers to a transformation that makes a mathematical object either bigger or smaller in some dimension, but does not move it’s center.
We will let $f(x)$ be represented by the map that defines our orange right triangle to the right. Transformations of the right triangle will be shown in blue while the original right triangle will remain in orange for the sake of comparison.
Below, we will consider a variety of transforms. Notice my transforms have a certain order to them that is not commutative. I must first move horizontally h, then perform dilation (possibly horizontal reflection) b. Then I scale by a (and possibly vertical reflection) and move k vertically.
You may download the excel workbook and explore transformation on your own below.
I’d like to concentrate on the last example $-2f(\frac{-1}{2}(x-1))-2$. Let’s explore these transformations in order.
