Find a condition which determines whether the hyperbola y^2 - x^2 =
k contains any points with integer coordinates.
This task develops knowledge of transformation of graphs. By
framing and asking questions a member of the team has to find out
which mathematical function they have chosen.
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 ?
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.
This is a beautiful result involving a parabola and parallels.
Here is a pattern composed of the graphs of 14 parabolas. Can you
find their equations?
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.
Alison has created two mappings. Can you figure out what they do?
What questions do they prompt you to ask?
Explore the two quadratic functions and find out how their graphs
Here are some more quadratic functions to explore. How are their
Explore the relationship between quadratic functions and their
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?
Take a few whole numbers away from a triangle number. If you know
the mean of the remaining numbers can you find the triangle number
and which numbers were removed?
An inequality involving integrals of squares of functions.
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?
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.