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Reflecting Lines

Age 11 to 14 Challenge Level:

Here are two sets of graphs. Each shows a pair of lines which are reflections of each other, one in the horizontal axis and one in the vertical axis.

screenshot of GeoGebra tool showing two lines reflected in the line y=0 screenshot of GeoGebra tool, showing two lines reflected in the line x=0. 

Move the red and blue dots on the interactivity below to create some more pairs of reflected lines.

What can you say about the equations of two lines if one is a reflection of the other in the horizontal axis? What about a reflection in the vertical axis?

Below are the equations of sixteen straight lines. Each line has a partner, either its reflection in the horizontal axis or its reflection in the vertical axis. Without plotting any graphs, can you find all the pairs and say which axis they were reflected in?

$y=4x+3$ $y=2x+4$ $y=2x+\frac14$

$y=-4x-2$ $y=\frac14x+2$ $y=-2x-4$

$y=3x+4$ $y=-3x-\frac14$ $y=\frac14x-3$

$y=-4x+3$ $y=4x+2$ $y=-3x+4$

Now imagine that a line is reflected in one of the axes and the image is then reflected in the other.

Can you predict the equation of the resulting line if you know the equation of the original?
Does it make a difference which axis you choose to reflect in first?
Explain your findings.