Find parametric equations and a parameter interval for the...

Find parametric equations and a parameter interval for the motion
of a particle that moves along the graph of $y=x^{2}$ in the following
way: Beginning at $(0,0)$ it moves to $(3,9),$ and then travels back
and forth from $(3,9)$ to $(-3,9)$ infinitely many times.

Key Concepts

Parameter Interval

The parameter interval defines the range over which the parameter, typically time, is allowed to vary in the motion description. Choosing appropriate intervals is crucial, especially when the motion involves different phases, such as an initial transit followed by an infinitely repeating cycle, to ensure that the behavior is captured accurately over time.

Piecewise and Periodic Motion

When a particle's motion consists of different segments—such as a one-way journey followed by an oscillatory back-and-forth pattern—the use of piecewise functions becomes important. Periodic motion captures the concept of repetitive movement over a defined interval, allowing the representation of cycles or repeated behavior in the particle’s path.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of an independent parameter, typically time. They allow one to represent both x and y coordinates separately, making it easier to describe complex motions and trajectories that may not be expressed as a single function in Cartesian form.

Motion along a Curve

Motion along a curve involves describing how a particle travels as a function of time along a defined path. This concept requires understanding the relationship between the parameter, often time, and the particle's position on the curve, which is essential for analyzing dynamics and trajectories.

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