Is there any way to find an estimated formula formula for the tangent line with this given information. y=f(x). f'(-3), f'(0) and f'(3)
well when you take f'(a) you get the slope of f(x) at that the point a.
On the other hand, you also are getting the slope of the tangent line at a.
So if you know f'(-3), f'(0), f'(3) you would know the slopes of the tangent lines at x = -3, x=0 and x=3.
However that doesn't guarantee that all the tangent lines are the same. If you had some extra information you could possibly do something, but unless the derivative is constant, you're looking at 3 different tangent lines.
(think about the tangent lines along a sine or cosine for example)