Given a graph of the displacement of a particle, how can you tell if it is in Simple Harmonic Motion?

There are two main features if an object is in SHM. The first is that it has a fixed maximum amplitude. That is to say, if the mean (halfway between the maximum values of oscillation (+ve and -ve)) displacement is taken as 0, then the maximum values of amplitude are both A and -A. So the total maximum displacement from the average position is always A. The second main feature is that the acceleration the particle undergoes is always proportional to the displacement from the mean/equilibrium position. This is an important criteria, as it limits the forms of motion that the particle can have. From looking at the basic equation of a SHM, x = Asin(ωt + φ) where x is the displacement of the particle, A is the amplitude (or maximum value of the displacement), ω is the angular frequency, and φ is for any phase offset. Therefore, the velocity for any given displacement is dx/dt = ωAcos(ωt + φ) And the acceleration is d²x/dt² = - ω²Asin(ωt + φ) which can also be expressed as a = - ω²x which satisfies that the acceleration, a, must be proportional to x. And so, from this, it is obvious that a particle undergoing SHM must be sinusoidal on a displacement-time graph, in order for it to meet both criteria for being in SHM. If you are able to draw the derivatives on the graph you have been given, you will notice that the acceleration is π radians out of phase of the motion.