Elliot from Wilson's School sent us the
The first two functions are almost identical, as they both square
the number, add on the original number and then subtract six.
However, function $g$ also adds two at the end, causing function
$f$ to always be two less than function $g$, when $x$ is the same
number. Because of this, their graphs are almost identical curves,
apart from function $f$'s graph being two below function $g$'s
The graph of function $f$ dips to $-6.25$ and function $g$ dips
down to $-4.25$.
Krystof from Uhelny Trh, Prague, describes
this by saying that the second function "shifts" the first function
upwards by 2 units.
Elliot goes on to explore the second
For the second pair of functions, function $f$ remains the same,
but function $g$ now adds on two before
the number is squared,
instead of at the end. This changes the relationship considerably,
as before, the output of $f$ would always be two less than function
$g$'s output, but if the number is squared after
two is added, the
difference between the outputs is very different. The two graphs
are again very similar to each other, however, this time function
$f$ produces a curve slightly to the right of function $g$'s curve
with them both dipping down to $-2.5$.
In fact, function $g$'s curve is two units
to the right of function $f$'s curve - can you see
George, who is also from Wilson's School,
sent us a very clear explanation of the first part of the problem,
which you can read by clicking here.