Quadratic functions and 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

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

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

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

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.
problem

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?