# Parabolic Angry Birds

Recently, our math teacher gave us a project and in it, it asked Blue Bird began at the location (6,0) and hit the ground at (26,0). Write a polynomial equation that could represent his flight path and draw its graph. I was able to figure out that the axis of symmetry was 16 and am one step closer to getting it into vertex form. The question that I have is how do you get vertex form from two points?

• You can't - there are infinitely many quadratic curves with those two roots that are symmetric about $16$ (just imagine one such curve and stretch it about in the y-direction). However, given a certain acceleration (say 9.81) you'd have a unique curve. Also, the question your teacher set is very carefully worded - it says to write a polynomial. So I guess any polynomial (quadratic) through those points is a valid path. – John Doe Apr 11 '18 at 1:32
• Hint: how many roots does a quadratic have? – John Douma Apr 11 '18 at 1:32
• How does physics work in the world of angry birds? – Bram28 Apr 11 '18 at 1:33
• At face value, and not knowing/assuming anything about birds or physics, the problem only states that $\,p(6)=p(26)=0\,$ for some polynomial $\,p\,$. Then the general solution is $\,p(x)=(x-6)(x-26)q(x)\,$ where $\,q\,$ is an arbitrary polynomial. This includes several quadratics, with some that go underground, and also the zero polynomial $\,p(x) \equiv 0\,$ in case the bird chooses to walk. – dxiv Apr 11 '18 at 3:50