Our general equation for position in SHM is: x(t) = A cos(ωt + φ)

Let's analyze each piece of this equation separately.

Change the Oscillation Amplitude: For SHM the oscillation frequency is amplitude independent!

Change the Oscillation Frequency: For a spring-mass system, frequency is changed by changing k/m, because ω = (k/m)^½. If we keep k the same but double the mass, the frequency is reduced by a factor of the square root of two.

Change the Phase: If all that is different is the phase, the graphs of x(t) are identical except that one is shifted relative to the other by the phase angle.