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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

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Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

Quadratic Rotation

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

$x^{2} - 6x + 11 = (x - 3)^{2} + 2$.

When the curve is rotated $180^\circ$ about the origin, the equation of the new curve will be
$y = -(x + 3)^{2} - 2$
   $= -x^{2} - 6x - 9 - 2$
   $= -x^{2} - 6x - 11$.

Note: the image of point $(a,b)$ under a $180^\circ$ rotation about the origin is the point $(-a,-b)$. An alternative method, therefore, is to replace $x$ and $y$ in the original equation by $-x$ and $-y$ respectively.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.