Another plot, another challenge:

A bit more of a traditional plot, with several good answers all along the same lines:

  • @OneDavidAtATime on Twitter was (I think) first off the mark: “I see the same function twice, which could be… a ratio? Something like $y=\frac{kx}{(a-x)(a+x)}$, $a$ and $k$ parameters. The red area would be the desired interval for the input $x$ so the output ratio is $y_1 \lt y \lt y_2$. $x$ could be some type of spatial shift around point $a$.”
  • @pretentious7 on Mastodon came in at about the same time with: “Hmmm looks sort of like a sideways logistic with a hyperbolic split.”
  • And my good friend Matthew on LinkedIn gave the fullest answer:

“Hello. Your mysetery plot - I was going to step thorugh the logic:

  • Area between two curves : $f(x) < y < g(x)$ (although this isn’t consistently true, in places it’s $g(x)< y < f(x)$ - just spotted that)
  • two vertical asymptotes each (which I’m pretending are -2, 2 and $-\frac{1}{2}$, $\frac{1}{2}$: $f(1/2)=0$, $f(-1/2)=0$, $g(2)=0$, $g(-2)=0$
  • put the zeros as factors in the denominator -> crosses the origin with positive slope => numerator is of the order of $x$

so $f(x) = -\frac{x}{(x-1/2)(x+1/2)}$ and $g(x) = -\frac{x}{(x-2)(x+2)}$ but then it isn’t the area between these lines in the usual manner….

(As an aside, I point at a couple of Matthew’s mathematical problem-solving techniques: one is to make a note of possible problems, like the difference in order of the functions; another is to pick easier numbers than the numbers in the actual problem and look at the structure of the graph rather than the details. He’s a clever chap.)

Postcards for all three!

For the record, the actual function I plotted was $\left( x^2 + \frac{x}{y} - 3\right)^2 \le 8$.

Stay tuned for the next unmissable episode of What’s The Plot, probably in a few weeks’ time!