Position of a particle from the graph of its velocity

[Tarrok]

Homework Statement

Problem: how do we know for sure if x is <0 for point A?

V_x-t is a graph of particle's speed over time.
x represents position of the particle at any given time

The Attempt at a Solution

From the function that we can see on the left side of the picture we can deduce the formula for it. It should be something like v = -a(t-b)^2+c, where a,b and c are some unknown constants >0.

So i attempted to use wolframalpha to plot a function similar to this:

And then I took integral of this function to find the connection between x (position of the particle) and t:

From this function we can see that if the constant is equal to zero, then position x of the particle at t=0 should be zero. This means that since the textbook says that at point A x<0, our constant is =/=0, but instead is a negative number.

But how do they know that constant is <0 at point A?
All we have is a graph of the derivate of the function of particle's position (x) over time. When we take a derivative of this, the constant turns into zero, so I don't see the way that authors used to determine whether the constant is >0, <0 or 0. Why do they state that x<0 at A then if it could actually be anything?

Thanks for help!

 

[PeroK]

A graph of velocity against time does not imply an initial position. The book must have stated or assumed that $x(t=0) < 0$.