Transformations of Quadratic Functions

We want to describe how to transform the parent function $f(x)=x^2$ to the graph of the given quadratic function. $$\begin{array}{c}g(x)=(x-9{)}^2+5\end{array}$$ To do so, we need to consider two possible transformations.

1. Horizontal translations
2. Vertical translations

Let's consider them one at the time.

Horizontal translation

If an addition or subtraction is applied to only the $x$-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, $9$ is being subtracted from $x,$ so the previous graph will be translated $9$ units to the right.

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, $5$ is added to the whole function, so the previous graph will be translated $5$ units up.

Final graph

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

Finally, let's summarize all the transformations of the graph of $f.$

• A horizontal translation $9$ units to the right
• A vertical translation $5$ units up