Graph sketching

problem
Witch of Agnesi

Age

16 to 18

Challenge level

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.
problem
Maltese cross

Age

16 to 18

Challenge level

Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.
problem
Folium of Descartes

Age

16 to 18

Challenge level

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.
problem
Cocked hat

Age

16 to 18

Challenge level

Sketch the graphs for this implicitly defined family of functions.
problem
Sine problem

Age

16 to 18

Challenge level

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.
problem
Power up

Age

16 to 18

Challenge level

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x
problem
Quartics

Age

16 to 18

Challenge level

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.
problem
How many solutions?

Age

16 to 18

Challenge level

Find all the solutions to the this equation.