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Some good thinking from Berny at
Gordonstoun School, and others, connecting the algebra to the shape
and position of the quadratic graph (parabola) :
When we set a problem on the NRICH site there's often an insight we
hope you'll discover as you work your way around the problem.
In
Minus One Two Three the
insight was to see that expressions a. b. and c. below are just
three different ways to represent the same quadratic form.
b. is made from a., multiplying to 'remove the brackets', and
c. is decided by looking at b.
Deciding the c. form is sometimes called 'completing the
square', if you are not yet familiar with that, square $( x + 4 )$
to check that the three representations really do match.
It's, b, the second of those three forms, which helps most here
because it reveals the axis of symmetry for the red graph below (
the blue graph is a simple x-squared parabola passing through the
origin )
Looking at c. we can see that $x = -4$ will produce the lowest
possible value for this expression ( $-4$ ).
The '$+ 4$' in the expression works like a boost for the $x$ value
and lets everything on the $x$-squared curve happen $4$ 'earlier' -
that is to the left of the origin.
Subtracting $4$, after the 'squaring', has the effect of lowering
all values, or points on the graph, by $4$.
So c. is like the $x$-squared parabola but shifted four left and
four down.
For x smaller than $-4$ the graph will be 'falling' towards this
minimum value and to the right (greater than $-4$) the graph will
'climb' again.
So as x values move from $-3$ , to $-2$ , to $-1$ , the value of
the expression will be increasing, away to the right of that
minimum point of $x = -4$.
So $-3$ will produce the lowest of these three values, and $-1$
will produce the highest.
In a similar way with the other expressions :
$(2x + 7)(x + 1)$ produces middle, lowest, and highest of the three
values when $x = -3$, $-2$, and $-1$
$(2x - 3)(x + 5)$ produces highest, lowest, and middle of the three
values when $x = -3$, $-2$, and $-1$
$(x - 3)(x - 1)$ produces highest, middle, and lowest of the three
values when $x = -3$, $-2$, and $-1$
More generally, you might like to think
about whether every pair of linear brackets, like a. would produce
a parabola-type ($x$-squared) curve, and similarly whether every
expression like b. would have to be symmetric about some $x$
value.