How to roughly find the function of the following graph?

Question

Given a graph of function as follows.

• $f(0)=2$
• $f(\pm 1)=0$
• $f(\pm 2)=-1$ is the local minimum value
• $f(x)\to 0$ when $x\to \pm \infty$
• $f(\pm 4)\approx -10\%=-0.1$

Attempt

With the help of Wolfram Mathematica, I estimated the function as follows.

$$ f(x)=\frac{16 \left(1-x^2\right)}{ x^4 \sqrt{2} +2\left(5-2 \sqrt{2}\right) x^2+8} $$

It is hard to adjust the $f(\pm 4)\approx -10\%$ and the local minimum.

Question

How to roughly find the function of the following graph?

Answer 1

You can take $$ f(x)=\frac{2(1-x^2)}{1+a x^2+bx^4+cx^6} $$ and the conditions lead to the values: $$ a=\frac{35}{16},\quad b=-\frac{21}{32},\quad c=\frac{27}{256} $$ so simplifying $$ f(x)=\frac{512(1-x^2)}{256+560x^2-168x^4+27x^6} $$