Summary

x=A\cos(2\pi ft) for when object starts at max displacement (cos(0) = 1)
x=A\sin(2\pi ft) for when object starts at equilibrium (sin(0) = 0)
Velocity is largest at equilibrium, acceleration is largest at turning points.
Amplitude does not affect period.
Velocity time graphs are shifted by a quarter period from displacement.
Acceleration time graphs are shifted by half a period.
Reading zeros and extrema tells position, direction of motion, and whether the object is speeding up or slowing down.