Purpose of coefficients in a equation or expression?

[pairofstrings]

Hello.
I have an equation which I understand partially but after understanding the purpose of coefficients in the below equation I think my understanding will enhance.

* I found it on: https://www.desmos.com/calculator/dnzfajfpym

What decides that I need coefficients for x² in the given equation of this post?

I think that coefficients of x² in this post's equation exist to scale up or scale down the size of the thing drawn in blue box: Correct?

But how do I come up a number that is so precise like:

which is the coefficient of x² to complete the sketching of the curve (Batman)?

Do I need to solve a few equations to come up with the coefficient of x² in the following equation?

Thanks!

 

[mfb]

Do I need to solve a few equations to come up with the coefficient of x² in the following equation?

Just one. It should match the elliptic parts outside: At $x = \pm 4$ the value of the function has to be about -2.45 to make the curve continuous. You can get the exact value from curve 2: It has to be $-3\sqrt{1-(4/7)^2}$