problem
Enclosing squares
Age
11 to 14
Challenge level
Can you find sets of sloping lines that enclose a square?
problem
Parabolic patterns
Age
14 to 18
Challenge level
The illustration shows the graphs of fifteen functions. Two of them have equations $y=x^2$ and $y=-(x-4)^2$. Find the equations of all the other graphs.
problem
More parabolic patterns
Age
14 to 18
Challenge level
The illustration shows the graphs of twelve functions. Three of
them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations
of all the other graphs.
problem
Ellipses
Age
14 to 18
Challenge level
Here is a pattern for you to experiment with using graph drawing
software. Find the equations of the graphs in the pattern.
problem
Parabolas again
Age
14 to 18
Challenge level
Here is a pattern composed of the graphs of 14 parabolas. Can you
find their equations?
problem
Exploring cubic functions
Age
14 to 18
Challenge level
Quadratic graphs are very familiar, but what patterns can you explore with cubics?
problem
Climbing
Age
16 to 18
Challenge level
Sketch the graphs of y = sin x and y = tan x and some straight
lines. Prove some inequalities.
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
Pitchfork
Age
16 to 18
Challenge level
Plot the graph of x^y = y^x in the first quadrant and explain its
properties.
problem
Limiting probabilities
Age
16 to 18
Challenge level
Given probabilities of taking paths in a graph from each node, use
matrix multiplication to find the probability of going from one
vertex to another in 2 stages, or 3, or 4 or even 100.
problem
A close match
Age
16 to 18
Challenge level
Can you massage the parameters of these curves to make them match as closely as possible?