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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?

More Parabolic Patterns

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

Sketch the graphof $y=x^2$. If you exchange $x$ and $y$ the point $(a,b)$ on one graph has an image $(b,a)$ on the other graph. What transformation of the graphs will have this effect? What do you expect the graph of $x= y^2$ to look like? Find these two graphs on the illustration.

Now use a graphic calculator, or graphing software, to sketch the two graphs. Were your predictions correct?

Find the other equations by considering the transformations of the graphs, changing the equations accordingly and testing your decisions by sketching the graphs and seeing if they match the graphs in the illustration in the way that you expected.