In this section, you will use an interactive sketch to investigate translations. Click on the following link to open the applet in a new tab/window.

Transmographer

This sketch allows you to investigate the changes in coordinates when a reflection is applied to a geometric figure.

Reflections

Now, let’s investigate reflections across the y-axis.

1. Use your notes to record the initial coordinates of the green point, yellow point, cyan point, and black point.
2. In the Reflect box, check “across x = 0”. A reflection across the line x = 0 is really a reflection across the y-axis.
3. Click the Reflect button.
4. Record the new coordinates of each point.
5. How did the value of the coordinates change?
   Interactive popup. Assistance may be required.

   Check Your Answer

   The y-coordinate remains the same, but the x-coordinate is transformed into its opposite.
6. How was the size and shape of the figure affected?
   Interactive popup. Assistance may be required.

   Check Your Answer

   The size and shape did not change.
7. Repeat this process using different polygons.
8. In general, how are the x and y-coordinates affected when a figure is reflected across the y-axis?
   Interactive popup. Assistance may be required.

   Check Your Answer

   The y-coordinate remains the same, but the x-coordinate is transformed into its opposite.

We can write the rule for a reflection across the y-axis in symbolic language.

P(x, y) → P^t(−x, y) or r_y−axis (x, y) = (−x, y)

In general, how are the size and shape of a figure affected when a figure is reflected across the y-axis?

Interactive popup. Assistance may be required.

Check Your Answer

The size and shape did not change.

Since the size and shape of figures are not affected when a figure is reflected across the y-axis, we say reflections across the y-axis are congruence transformations.