If I wanted to calculate the force constant and mass of a harmonic oscillator given an energy vs. time graph, how would I do it?

Here's what I would do.

You should be given the equilibrium spring length if you are to do this, or you can find it along the #x# axis, presumably, if you also have a spring length vs. energy graph (along with a time vs. energy graph).

I derive below to get

#k = (2overbrace((1/2 mv^2))^(K))/(r_e^2)# in #"kg/s"^2#

#m = (kT^2)/(4pi^2)# in #"kg"#

The equilibrium spring length can be found when the kinetic and potential energies balance out, i.e. at their intersection here:

So, let's say you knew the kinetic energy at the intersection. Then:

#overbrace(1/2 kr_e^2)^(U) = overbrace(1/2 mv^2)^(K) = "??? kg"cdot"m"^2"/s"^2#

#=> color(blue)(k = (2K)/(r_e^2))#

where #r_e# is the equilibrium spring length, whereas #A# would have been the maximum spring length. I would hope you are given #r_e# or you have a spring length vs. energy graph.

Then, one can use the relation #omega = sqrt(k/m)#, where #omega# is the angular frequency in #"rad/s"#, #k# is the force constant in #"kg/s"^2#, and #m# is the mass in #"kg"#. Thus,

#omega^2 = k/m#

#=> m = k/(omega^2)#

and the angular frequency can be found from the period:

#omega = 2pi f#, where #f# is the frequency in #"s"^(-1)#,

while the period is #T = 1/f#, so #omega# in terms of #T# is

#omega = (2pi)/T#

The period can be found from the graph by looking at the distance from crest/trough to crest/trough. So, this gives a final relation of:

#color(blue)(m) = k/((4pi^2)/T^2) = color(blue)((kT^2)/(4pi^2))#