Energy versus Position Curves

The potential energy stored in a spring is U(x) = (1/2)kx^2. That's just a parabola opening upward, centered at x = 0, because U depends on x squared: stretch or compress the spring by the same amount in either direction and you store the same energy. The bottom of the parabola, at x = 0, is equilibrium, the natural length of the spring. Force is the negative slope of the potential energy curve, and the slope at the very bottom of a parabola is zero, so the force there is zero. That's exactly why equilibrium is a resting point: nothing pushes or pulls the mass when it sits at x = 0.

Total mechanical energy, kinetic plus potential, stays constant through the motion, so on this graph it's a flat horizontal line. Where that line crosses the parabola, all the energy is potential and none is kinetic, meaning the mass has stopped and is about to turn around. Those crossing points ARE the turning points, and the distance from x = 0 out to either one is the amplitude. At any position in between, kinetic energy is the vertical gap between the horizontal total-energy line and the parabola directly below it: KE = E_total - U(x). That gap is biggest at x = 0, where the mass moves fastest, and shrinks to zero right at the turning points.

Give the oscillator more energy and the horizontal line sits higher. A higher line hits the same parabola at points farther from the center, so the amplitude grows. This is the same energy-versus-position reasoning from the work and energy unit, just applied to a specific U(x) shape: the curve's bottom is stable equilibrium, the slope gives force everywhere, and the vertical gap below any horizontal energy line is always kinetic energy.