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This problem challenges you to find cubic equations which satisfy different conditions. You may like to use Desmos to help you investigate possible cubics.
Part 1
Can you find a cubic which passes through $(0,0)$ and the points $(1, 2)$ and $(2,1)$?
Can you find more than one possible cubic?
Part 2 (a)
Can you find a cubic which passes through $(0,0)$ and the points $(1, 2)$ and $(2,1)$, and where the point $(1,2)$ is a turning point of the cubic?
Can you find more than one cubic satisfying all the conditions?
Part 2 (b)
Can you find a cubic which passes through $(0,0)$ and the points $(1, 2)$ and $(2,1)$, and where the point $(2,1)$ is a turning point of the cubic?
Can you find more than one cubic satisfying all the conditions?
Part 3
Can you find a cubic which passes through $(0,0)$ and where the points $(1, 2)$ and $(2,1)$ are both turning points?