HOW TO DO THIS WITH A GRAPH?

Question

Rectangle PQRS has vertices P(1, 4), Q(6, 4), R(6, 1), and S(1, 1). Without graphing, find the new coordinates of the vertices of the rectangle after a reflection over the x-axis and then another reflection over the y-axis. (SHOW WORK)

Answer 1

When reflecting the rectangle on the x-axis, your points closest to the x-axis do not change, so those points will remain the same. Likewise, when reflecting about the y-axis the points closest to the y-axis will not change.

To find your new points, we first need to find how wide and tall your rectangle is by determining the differences.

Finding the differences on points for the rectangle's height (how far it spans across the y-axis) will require you to find the differences in Y on two points. We only need to use two points because your rectangle has the same X and Y value for two separate groups of data points, meaning it is lying flat instead of being drawn at an angle.

Using the points P and S we can find the difference in height or across the y-axis. I chose these points because they have the same x-values.

P is (1,4) while S is (1,1)

the difference in height, or y-values, is found by subtracting both y-values 4 - 1 = 3

You could also subtract 1 - 4 then convert it to a positive number, but I find it easier to subtract into positive numbers when possible.

Next we will need to find the difference in x-values, or what we would often consider width. To do this we will do the same thing as finding the y difference, but with points P and Q (you could also use P and R, S and Q, or S and R because each pair shares the same Y-value).

P is (1,4) while Q is (6,4) so they have the same Y-value but different x-values.

To find the difference in x-values we will subtract 6 - 1 = 5

To reflect about the x-axis, we will want to use the points closest to the x-axis, R and S (6,1) and (1,1). We know they are closest because they have the lowest y-values.

From here, we can go 3 points down from each point so the rectangle will remain the same size. This will give us points (6,-2) and (1,-2) as 1 - 3 = -2

Reflecting about the y-axis will be similar, but we will use points closest to the y-axis (having the lowest x-values) and extend the distance of the x-value difference found previously.

The point most left are P and S (1,4) and (1,1)

From here we will go left 5 points, the difference in x-values we found previously.

1 - 5 = -4 so our new points will be (-4,4) and (-4,1)

I hope this was helpful in your understanding of reflections and examining shaes without the use of graphs.