Potential energy curve

Plot a system's potential energy against position and the graph hands you the force, the equilibrium points and the range of the motion before you solve anything.

Slope gives force. The AP Physics C: Mechanics sheet prints $F_x = -dU(x)/dx$, and the CED supplies the sign in words: these forces point in the direction of decreasing potential energy. Downhill on the graph is the way the force pushes.

Flat spots are equilibrium positions, and the curvature says which kind.

• A local minimum is stable equilibrium. A small displacement produces a force opposite to that displacement, accelerating the object back toward the equilibrium position, so the system oscillates about it.
• A local maximum is unstable equilibrium. A small displacement produces a force in the same direction as the displacement, accelerating the object away.

Turning points come from drawing the total energy as a horizontal line across the graph. Since $K = E - U$, the kinetic energy is the vertical gap between the line and the curve. Where they meet, $K = 0$ and the motion reverses, and any region where the curve rises above the line cannot be reached at that energy. That is how the picture tells you whether a system is trapped in a well or free to escape.

The slope relationship is calculus, so it belongs to AP Physics C: Mechanics. The algebra based sheets do not print $F_x = -dU(x)/dx$.