### 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.

### Cubics

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

Can you adjust the curve so the bead drops with near constant vertical velocity?

##### Stage: 4 Short Challenge Level:

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

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.
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