solving cubic equations iteratively

a method of solving a cubic equation that can be fraught with difficulties (smoothed out in trial GCSE questions)

there are much better ways to solve a quadratic equation but an iterative approach works quite well

perhaps we could teach the general cubic solution (and quartic equation solution)??

self-inverse and periodic functions

some functions, if repeatedly used (i.e. iterated) return to a start value and it can be interesting to explore these - by operating with fractions preferably, or using a calculator

it can be demanding to firmly establish how many steps it takes to return to a start value (the period) algebraically

the 'usual' self-inverse functions are of a form similar to:
f(x) = 10 - x and
f(x) = 12/x

what happens if you work out 10 - 3 and then subtract the result from 10?
leading to an appreciation that 10 - (10 - n) is n

however, there are many more examples of self-inverse rational functions
here are a few:

etc.
etc.
there is a simple enough rule for such functions to be self-inverse

but it's not that easy to justify...

for practice in dividing fractions and working with directed numbers, self-inverse functions provide an interesting context:

• put a number ('seed' value) into one of the functions (above) 
• feed the output (y-value) back into the function

what happens?
try this with several input (start) numbers...


why is the process problematic if the denominator is = 0?
other than for a problematic x value, does the number always seem to return to the start?
what if x is a fraction, or a negative number, or a negative fraction?

what happens with these functions?

destination douze

introducing iterative sequences
with a calculator and using the 'ans' key to take forward an answer

[e.g. for question 1 with a seed value of 6:
enter 6, then =
then ans x 2 /3 + 4
=
=
=
a lot of times... 52 to be precise]

students put in any number (a 'seed' value)
and then apply a rule repeatedly - with the outcome becoming the new input etc.
until they have a good reason to stop

what happens for these three iterative rules?
what is the limit?
does it matter what the seed value is?

other iterative sequences have the same limit
what (constant) value needs to be added?

can other iterative sequences with this limit be found?
e.g. for a fraction of 3/8?