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Reflecting Lines
Age 11 to 14
Challenge Level 
Jiwon Jung noticed:
The lines that are reflections have the same absolute value of their x coefficient and the constant.
Well noticed. Ellie from Chiswick matched up the reflections, and explained why they work:
A straight line is $y=ax+b$, where $a$ and $b$ are constants,
and can be positive or negative.
If you reflect in the $x$ axis, then all the $y$ values become
$-y$, so the line becomes $-y=ax+b$, which we write
$y=-ax-b$.
If we reflect in the $y$ axis, then the $x$ values become
$-x$, and the line becomes $y=a(-x)+b$, which is $y=-ax+b$.
Now we can match the lines.
Then I looked at doing both reflections.
First, I tried reflect in $x$, then $y$, using the rules
above.
$y=ax+b$ reflected in $x$ is $y=-ax-b$, then reflected in $y$
becomes $y=ax-b$
Now try the other way around.
$y=ax+b$ reflected in $y$ is $y=-ax+b$, then reflected in $x$
becomes $y=ax-b$
So it doesn't matter which way round you do the reflections.
The twice reflected graph has the same gradient as the original,
but translated down so it crosses the $y$ axis at the opposite
place.
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