Parabolic Patterns
The illustration shows the graphs of fifteen functions. Two of
them have equations
$y = x^2$
$y = - (x - 4)^2$
Use a graphic calculator or a graph drawing computer program to
sketch these two graphs and then locate them in this illustration.
Use the clues given in this information to help you to find the
equations of all the other graphs and to draw the pattern of the 15
graphs for yourself. For your solution send in the equations you
have found with an explanation of how you did it.
What about the equations of these parabolas?
You may like to use your creative talents to devise your own
pattern of graphs and send them to us so that we can base another
challenge like this one on the website using your pattern.
NOTES AND BACKGROUND
This sort of challenge is sometimes called an inverse problem
because the question is posed the opposite way round to what might
have been expected. This is almost like saying: 'here is the
answer, what was the question?' Instead of giving the equations of
some functions and asking you to sketch the graphs, this challenge
gives the graphs and asks you to find their equations.
You are being asked to sketch a family of graphs. What makes this a
family? All the graphs are obtained by transformations such as
reflections and translations of other graphs in the family. The key
is to find the simplest function and then tofind transformations of
the graph of that function which give the other graphs in the
family.
If you have access to a graphic calculator, or tograph drawing
software, it will not give you the answers. You will have to think
for yourself what the equations should be and then the software
will enable you to test your own theories and see if you were
right.