"A ball is thrown upward from the roof of a building $60$m tall. The ball reaches a height of $80$m above the ground after $2$ s and hits the ground $6$ s after being thrown." How do I determine the equation without using physics(constants related to physics) ideas?
It should be in the form $y= ax^2+ bx + c$, right? I got $60$ for the $y$-intercept so $y= ax^2+ bx + 60$. The ball had the height of $80$ meters at $2$ seconds. It dropped to the ground in $4$ seconds. So $\frac{80}{4}=b\implies b=20$? I found $a$ and got $-5x^2 + 20x + 60$ but do my steps show I understand the problem?
Physics says two things. First, that the acceleration from gravity is constant, and second, that the acceleration is $-9.81 m/s^2$. You have accepted the first, which results in $y=ax^2+bx+c$ (assuming $x$ is time) but rejected the second. That is fine, you have enough data to evaluate all the constants, as you have three points on the parabola and need three constants, $a,b,c$. As you say, $c=60$ is immediate.