I need help with finding equation. I tried to write parabola equation but it doesn't fit so much. I think that hyperbola will fit perfect. I have got points:
$$\left(0,\frac{\pi ^2}{6}-1\right),\left(\frac{1}{2},\frac{2}{3}\right),\left(\frac{3}{4},\frac{2\pi}{3}-\frac{88}{63}\right),\left(1,\frac{3}{4}\right)$$ and the same values for negative arguments, i. e.
$$\left(-\frac{1}{2},\frac{2}{3}\right),\left(-\frac{3}{4},\frac{2\pi}{3}-\frac{88}{63}\right),\left(-1,\frac{3}{4}\right).$$
Thank you very much for help.
I tried plotting the points, looks like $\left(1,\dfrac34\right)$ is a bad data in the set of given points. Drawing the points, tells us the vertex is on the y-axis, that is $(h,k)=\left(0,\dfrac{\pi^2}6-1\right)$
Now, we use the standard equation of a parabola $$y=a(x-h)^2+k$$
We substitute in the vertex. $$y=ax^2+\dfrac{\pi^2}6-1$$
Let's use the point $\left(\dfrac12,\dfrac23\right)$
$$\begin{align}\dfrac23&=\dfrac14a+\dfrac{\pi^2}6-1\\8&=3a+2\pi^2-12\\a&=\dfrac13(20-2\pi^2)\end{align}$$
Finally, the equation becomes $$y=\dfrac13(20-2\pi^2)x^2+\left(\dfrac{\pi^2}6-1\right)$$