Describing Transformations of Quadratic Functions

We want to describe how to transform the parent function $y=x^2$ to the graph of the given quadratic function. $\begin{gathered} y=\text{-}(2x)^2 \end{gathered}$ To do so, we need to consider two possible transformations.

1. Horizontal stretches and shrinks
2. Reflections

Let's consider them one at the time.

Horizontal stretch or shrink

We have a horizontal stretch when $x$ is multiplied by a number greater than one. If $x$ is multiplied by a number whose absolute value is less than one, a horizontal shrink will take place.

In the given exercise, $x$ is multiplied by $2.$ Therefore, the previous graph will be horizontally shrunk by a factor of $\frac{1}{2}.$

Reflection

Whenever $x^2$ is multiplied by a negative number, we will have a reflection of the graph across the $x$-axis.

Note how each $x$-coordinate stays the same, and how each $y$-coordinate changes its sign.

Final graph

Let's now graph the given function and the parent function $f(x)=x^2$ on the same coordinate grid.

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

• A horizontal shrink by a factor of $\frac{1}{2}$
• Reflection in the $x$-axis