Quadratic functions and graphs

problem
Favourite

Fence It

Age

11 to 14

Challenge level

1 out of 3

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
problem
Favourite

Minus One Two Three

Age

14 to 16

Challenge level

2 out of 3

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?
problem
Favourite

Parabolic Patterns

Age

14 to 18

Challenge level

1 out of 3

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
Favourite

Parabolas Again

Age

14 to 18

Challenge level

1 out of 3

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?
problem
Favourite

What's That Graph?

Age

14 to 18

Challenge level

2 out of 3

Can you work out which processes are represented by the graphs?
problem
Favourite

Parabella

Age

16 to 18

Challenge level

1 out of 3

This is a beautiful result involving a parabola and parallels.
problem

Grid Points on Hyperbolas

Age

16 to 18

Challenge level

1 out of 3

Find a condition which determines whether the hyperbola y^2 - x^2 = k contains any points with integer coordinates.
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More Parabolic Patterns

Age

14 to 18

Challenge level

1 out of 3

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

Janusz Asked

Age

16 to 18

Challenge level

2 out of 3

In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?
problem

Converse

Age

14 to 16

Challenge level

2 out of 3

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?