Determine a series of transformations that would map Figure X onto Figure Y.
Understand the Problem
The question asks us to identify the sequence of geometric transformations (such as translation, rotation, reflection, dilation, etc.) that, when applied to Figure X, will result in Figure Y.
Answer
Reflection about the x-axis, followed by a translation of $(-13, 5)$.
Answer for screen readers
Reflection about the x-axis, followed by a translation of $(-13, 5)$.
Steps to Solve
- Identify the necessary transformations
By observing the change of Figure X to Figure Y, it appears that a reflection and a translation are required. The orientation of Figure Y is flipped compared to Figure X, suggesting a reflection. Furthermore, Figure Y is located in a different quadrant than Figure X, indicating a translation.
- Determine the line of reflection
The shape of Figure Y suggests that Figure X is reflected about the x-axis.
- Determine the translation vector
After reflection about the x-axis, we need to translate Figure X to the location of Figure Y. Visually inspecting corresponding vertices after the reflection, it appears we need to translate the figure to the left and down. We can determine the translation vector by comparing coordinates of corresponding points.
For example the top right vertex of Figure X is at $(5, 9)$. After reflection it is at $(5, -9)$. The top right vertex of Figure Y is at $(-8, -4)$.
So the translation vector is $(-8 - 5, -4 - (-9)) = (-13, 5)$. This means we need to translate 13 units to the left and 5 units up after the reflection.
- State the transformations
Therefore, the sequence of transformations that maps Figure X onto Figure Y is a reflection about the x-axis, followed by a translation of $(-13, 5)$.
Reflection about the x-axis, followed by a translation of $(-13, 5)$.
More Information
Geometric transformations are operations that change the position, size, or shape of a geometric figure. Common geometric transformations include translations, rotations, reflections, and dilations. Identifying the correct sequence of transformations requires careful observation of the original and transformed figures.
Tips
A common mistake is to confuse the order of transformations. Performing the translation before the reflection, for example, will not map Figure X onto Figure Y.
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