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Can you find the maximum value of the curve defined by this expression?

Erratic Quadratic

Age 16 to 18 Challenge Level:

Find a quadratic graph which is at most 1 unit of distance away from each of the six points

(x, y) = (\pm 1, 0), (\pm 2, 4), (\pm 3, 16)

Can you find the smallest distance from each of these points to your quadratic, either exactly or to two decimal places?

Can you find another quadratic graph for which the largest of these smallest distances from the points to the curve is smaller?

As an open extension, explore what you can say about configurations of points at least one of which must necessarily lie more than one unit of distance away from a quadratic.