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.
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.
Explore the two quadratic functions and find out how their graphs
Here are some more quadratic functions to explore. How are their
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
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.
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?
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 ?
Find a condition which determines whether the hyperbola y^2 - x^2 =
k contains any points with integer coordinates.
Explore the relationship between quadratic functions and their
Alison has created two mappings. Can you figure out what they do?
What questions do they prompt you to ask?
This is a beautiful result involving a parabola and parallels.
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
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?
Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then
a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .
An inequality involving integrals of squares of functions.
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?