A curve-sketching masterclass | Colin

An implicit differentiation question dealt with $y^4 – 2x^2 + 8xy^2 + 9 = 0$. Differentiating it is easy enough for a competent A-level student – but what does the curve look like? That requires a bit more thought.

My usual approach to sketching a function uses a structure I call DATAS:
Domain: where is the function defined?
Axes: where does the function cross them?
Turning points: where are the function’s critical points?
Asymptotes: what happens when $x$ or $y$ get really big?
Shape: what does the graph look like overall?

As it turns out, this approach is almost entirely useless…

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