What was your method? Sometimes a "dodgy and inefficient method" is just because you lack confidence in what was actually a method that demonstrated a good understanding of the material.
For example, the way I would have approached this (if it was tech free), was to have a look at the
graph (without the dilation
), and then have a look at my first
two intersections with
:
.
The reason why I looked at the first two intersections is because these are the two intersections (as the question said) that the graph
makes with
, except they are squished or stretched by the dilation factor
. We know that the rightmost intersection at
is the second intersection, and this corresponds to the second intersection on the original graph
.
Now, the exact value of
falls out really easily now, because
.
From this, it follows that the first (left side) intersection is
I guess a less confident student would think this method is clumsy, but it is more elegant than clumsy, in my opinion.
The reason why this works is because the way the question is worded, it starts looking at the intersections from the point
. Since dilations sort of occur from
as the centre, you can use this method. The situation becomes a bit more difficult if your domain does not start at zero, because now a dilation factor would affect how many solutions occur before or after the domain (so you can't simply assume that the first two intersections from the original graph correspond to the first two intersections on the dilated graph).
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