An object's displacement in SHM can be represented by either of two equations, and which one you use depends entirely on where you decide t = 0 is:
Use cosine if the clock starts at the moment of maximum displacement (released from rest at x = +A or x = −A). Use sine if the clock starts at the moment the object passes through equilibrium. Both describe exactly the same kind of motion — they just start the story at a different point in the cycle.
A block on a spring starts at its equilibrium position and is given a push, so that at t = 0 it has its maximum speed. Which form, x(t) = A cos(2πft) or x(t) = A sin(2πft), correctly represents its displacement — and why?
You don't need new formulas to describe velocity and acceleration in SHM — you need to know where each one is at a maximum, a minimum, or zero, and why:
Recognizing these positions is what lets you qualitatively describe an entire cycle of motion without ever writing down v(t) or a(t) as formulas.
This isn't a new fact to memorize — it falls straight out of Lesson 7.1's SHM condition. The restoring force is proportional to −x, and Newton's second law says F = ma, so:
Acceleration is always proportional to displacement, with a negative sign built in. Whenever x is at a maximum, a is too — just pointing the opposite way. Whenever x is zero, a is zero. This is what "anti-phase" means: x and a always have opposite sign, everywhere in the cycle. Velocity, meanwhile, is 90° out of step with both — it peaks exactly where they're both zero, and vanishes exactly where they both peak.
Watch the cursor sweep across all three graphs together — one full pass is one period. Notice which graph is at zero exactly when another is at its peak.
A block on a spring starts at t = 0 at its maximum positive displacement (+A), released from rest. Describe the displacement, velocity, and acceleration of the block at t = 0, t = T/4, t = T/2, and t = 3T/4.
Properties of SHM — period, frequency, phase relationships — can be read directly off graphs of x, v, or a vs. time, without ever writing an equation. This is a skill worth practicing deliberately: given any one of the three graphs, you should be able to sketch the other two by reasoning through the extrema-and-zeros logic from this lesson.