Getting from a displacement vs$.$ time graph to an equation

A cart attached to a spring is displaced from equilibrium and then released. The graph

below shows the displacement as a function of time for the cart. There is no friction.

A$.$ Use the graph to figure out the period of the cart's motion, then use the period to figure out the frequency and

angular frequency of the oscillation.

Period $=4$s Frequency $=(1)/(4)$ Angular frequency $=(\backslash$pi $)/(2)$

B$.$ What is the mathematical expression for the displacement of the cart as a function of time? Keep the phase $\backslash$phi

in the expression.

x$($t$)=10$cos$((\backslash$pi $)/(2)$t$+\backslash$phi $)$

C$.$ Use the graph to figure out the phase, then write down displacement as a function of time again.

Hint: At what value of t is the displacement maximum in the graph? What must $\backslash$phi be in order for that to

happen?

Displacement is maximum at is $(10$cm$)$

D$.$ What are the mathematical expressions for the velocity and acceleration of the cart as a function of time?

E$.$ Based on your expressions above what are the maximum speed and the maximum acceleration of the cart?