Finding equations of transformed circles

Today, we started by going through the self-assessment exercises that were assigned last class. Some students had questions about the last problem. I went through the answers with the class and there were few questions. Even for the last problem, the questions were not major hindrances to them answering the questions.

I asked the class to indicate with one to five fingers, how well they did on these problems. A one would indicate they were totally lost to a five indicating they got all the problems correct. Almost the entire class held up three or four fingers. I had a couple of students who held up two fingers and none that held up one finger. So, the class seemed on firm footing with the exception of two students who continue to struggle a bit.

With that, I wanted to focus on transforming circles. I started with a circle of radius 5 centered at the origin. I asked the class a series of questions about different transformations and what would happen to the equation of the circle.

For example, we translate the circle using T(x + 2, y -3). Students were able to readily re-write the equation after translations. Next, I asked what the equation would be if the circle was reflected over the x-axis. This gave students some pause, but they soon realized they just needed to find the image of the center after the reflection. Reflecting over the y-axis then proved no problem. Reflection over the line y = x proved a bit more challenging, but students managed to determine the new center and the resulting equation.

Finally, I asked students what would happen if there was a dilation with a scale factor of 3. This baffled the class for a bit because the squaring of the radius was confusing them. After a while, some students started to realize that the radius changes and that the result of multiplying the radius and scale factor is what ends up being squared in the circle equation. To check things, I then asked what happens if the scale factor is 1/5?

At this point, I had students re-visit the four problems we had skipped on the first practice sheet of problems. These require completing the square. I asked students to tackle these. I don't expect them to get the correct answers, but by attempting the problems, they will have a better appreciation of what we will cover next class.