The Acceleration Function in One Dimension
Aside
Is there a function that describes how the acceleration is changing? Sure, you can find such a function by taking the derivative of the acceleration function. This function even has its own name, the jerk. While, there are specific applications that utilize the jerk and its jargon, they are relatively rare. The reason is that it is the accleration that is the connection between kinematics and forces. Newton's second law for the net force gives $$ \vec{F} = m\vec{a} $$ Typically, one wants the trajectory, or the position function of a particle. You find the forces on the object, which gives you the acceleration, from which you can find the position by integration.