Question

Please help me determine the general equation for the graph above as well as solve for a. Thank you.

Answer 1

Observe that the x coords of the roots of a polynomial are,

$x_{1,2,3,4}=\{-3,0,1,4\}$

Which can be put into form,

$y=a(x-x_1)(x-x_2)(x-x_3)(x-x_4)$

with data

$y=a(x-(-3))(x-0)(x-1)(x-4)=ax(x+3)(x-1)(x-4)$

Now if I take any root point and insert it into the equation I won't be able to solve for y because they will always multiply to zero (ie. when I pick $x=-3$ the right hand side will multiply to zero,

$y=-3a(-3+3)(-3-1)(-3-4)=0$

and a will be "lost" in the process.

If we observed a non-root point that we could substitute with x and y and result in a non-loss process then you could find a. But since there is no such point (I don't think you can read it of the graph) there is no other viable way to find a.

Hope this helps :)