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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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How Many Solutions?

Find all the solutions to the this equation.

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On the Road

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

Parabolas Again

Stage: 4 and 5 Challenge Level: Challenge Level:1

another family
Here is another family of graphs sent in by Andrei from Rumania together with his solution to this problem. Can you find the equations of the graphs in Andrei's illustration?

These challenges, together with the two challenges Parabolic Patterns and More Parabolic Patterns , develop the ideas of transformations of graphs and the corresponding equations.

The illustrations give families of graphs, all of parabolas, formed by reflections in the $x$ and $y$ axes and in the line $y=x$, together with translations and stretches parallel to the axes.

You are not expected to plot and join points although it is helpful to consider where the graphs cut the axes. Rather you should think of the features of the graphs and the effects of the transformations.