Describing Transformations of Quadratic Functions

We want to describe the transformation of the parent function $y=x^2$ represented by the quadratic function $$\begin{array}{c}y=(x+6{)}^2-2.\end{array}$$ To do so, we need to consider two possible transformations.

1. Horizontal translations.
2. Vertical translations.

Let's consider them one at a time.

Horizontal Translation

If an addition or subtraction is applied to only the $x\text{-}$variable, the graph will be horizontally translated. In case of addition, the graph will be translated to the left. In case of subtraction, it will be moved to the right. In the given equation, $6$ is being added to $x,$ so the graph of the parent function will be translated $6$ units to the left.

Vertical translation

If an addition or subtraction is applied to the whole function, the graph will be vertically translated. In the case of addition, the graph will be translated up. In the case of subtraction, it will be moved downwards. In the given equation, $2$ is subtracted from the whole function, so the previous graph will be translated $2$ units down.

Final graph

Let's now graph the given function $g$ and the parent function $f$ on the same coordinate plane.

We see that the graph of $g$ is a translation $6$ units to the left and $2$ units down of the graph of $f.$