Question 31

Question 31 A student made this conjecture about reflections on an xy coordinate plane: When a polygon is reflected over the y-axis, the x-coordinates of the corresponding vertices of the polygon and its image are opposite, but the y-coordinates are the same. The students should not use a calculator for this problem. The students may use a graph to prove that their answer is correct or if it is incorrect. Important things to keep in mind Y-axis: vertical axis X-axis: horizontal axis X-coordinate: first number in a point (ordered pair) Y-coordinate: second number in a point (ordered pair)

Question 31 Develop a chain of reasoning to justify or refute the conjecture. You must determine that the conjecture is always true or that there is at least one example in which the conjecture is not true. You may include one or more graphs in your reasoning. Here, the students are being asked to figure out if the previous statement is true or if it is false (it’s true). One thing the students will need to make sure they are doing is reflecting over the y-axis and not the x-axis.

Question 31 When a polygon is reflected over the y-axis, each vertex of the reflected polygon will end up on the opposite side of the y-axis but the same distance from the y-axis. So, the x-coordinates will change from positive to negative or negative to positive. This will make the x-coordinates of the corresponding vertices opposites. Since the polygon is being reflected over the y-axis, the image is in a different place horizontally but it does not move up or down, which means the y-coordinates of the vertices of the image will be the same as the y-coordinates of the corresponding vertices of the original polygon. This answer is taken almost verbatim from the answer guide. Pretty much it is saying that since it is just a horizontal switch and the shape is not moving at all, the shape will maintain the same y-coordinate but have opposite x-coordinates, since it is now in a different plane/quadrant. On the next slide is an example of a graphed example to see how it moved.

Question 31 (-6, 4) (6, 4) (-3, 2) (3, 2) (-8, 1) (8, 1) On this slide is a graph of what is happening to further prove that the conjecture is correct. It is the same one from the answer guide. The first triangle is at points (-8, 1), (-6, 4), and (-3, 2) The first thing that is shown is that (-8, 1) is 8 spaces away from the y-axis. This means that it will need to go 8 spaces away from the y-axis on the other side. In doing this, the height does not change, but the x-coordinate becomes the opposite sign. The second thing that is shown is that (-6, 4) is being reflected over the y-axis. Currently, it is 6 spaces away from the y-axis. We will have to move is 6 spaces in the other direction, as well. Like in the previous point, the line does not change in its height, so it is at (6, 4). This still makes the conjecture true. The last thing that is shown is that (-3, 2) is being reflected over the y-axis. Since it is currently 3 spaces away, so we move it 3 spaces past the y-axis. Like the other two points, it doesn’t change in height so the conjecture is true.