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"...or simply don't think of a graph to solve this because we think like math teachers and graphing has always been difficult"

to me this feels like the important part here. Graphing software fundamentally changes what is complex and what isn't. Before the software to quickly draw a graph and have it tell you where the minimum is we would either:

Sketch; which isn't useful when looking for an accurate answer

Draw accurately; which requires us to know the minimum already

But with graphing software the problem is no longer 'what's the minimum?' or 'what's the intersection?'. The problem is 'what do I need to do to make it possible to find the minimum?'

Is it cultural? Using graphs was hard in the past, so we shouldn't let students take the easy option?

to me this feels like the important part here. Graphing software fundamentally changes what is complex and what isn't. Before the software to quickly draw a graph and have it tell you where the minimum is we would either:

Sketch; which isn't useful when looking for an accurate answer

Draw accurately; which requires us to know the minimum already

But with graphing software the problem is no longer 'what's the minimum?' or 'what's the intersection?'. The problem is 'what do I need to do to make it possible to find the minimum?'

Is it cultural? Using graphs was hard in the past, so we shouldn't let students take the easy option?

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