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Describing Transformations of Quadratic Functions

Describing Transformations of Quadratic Functions 1.13 - Solution

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We want to describe how to transform the parent function y=x2y=x^2 to the graph of the given quadratic function. y=-(2x)2\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 xx is multiplied by a number greater than one. If xx is multiplied by a number whose absolute value is less than one, a horizontal shrink will take place.
Parent Function\text{Parent Function}

Shrink\text{Shrink}

Stretch\text{Stretch}

In the given exercise, xx is multiplied by 2.2. Therefore, the previous graph will be horizontally shrunk by a factor of 12.\frac{1}{2}.

Reflection

Whenever x2x^2 is multiplied by a negative number, we will have a reflection of the graph across the xx-axis.

Note how each xx-coordinate stays the same, and how each yy-coordinate changes its sign.

Final graph

Let's now graph the given function and the parent function f(x)=x2f(x)=x^2 on the same coordinate grid.

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

  • A horizontal shrink by a factor of 12\frac{1}{2}
  • Reflection in the xx-axis