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Erratic Quadratic

Can you find a quadratic equation which passes close to these points?


Age 16 to 18
Challenge Level

$X(r)$ is defined implicitly by the quadratic relationship
Part 1: Which of the choices $r=1,-1,100$ give real values for $X(r)$?

Part 2: What is the range of values of $r$ for which $X(r)$ takes real values?
What happens when $r=0$?

Part 3: Sketch the overall shape of $X(r)$ against $r$ and find the maximum and minimum values of $X(r)$.

Note: You could numerically find a sensible conjecture for the minimum and maximum values of $X(r)$, but to prove this you will need to use calculus.