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When working with parametric equations, like those given in problems describing the motion of a particle, it's often useful to convert these into a Cartesian equation. This equation describes the path of the particle in the familiar Cartesian coordinate system.
The Cartesian equation allows you to express the relationship between the coordinates without directly involving the parameter, usually denoted by \(t\). In this context, the Cartesian equation we derived is \(y = 1 - 2x^2\). This equation tells us that for every \(x\), there is a corresponding \(y\), thereby eliminating \(t\) from the picture.
Converting parametric equations to a Cartesian equation usually involves:

• Identifying the parametric equations, for instance, \(x = \sin t\) and \(y = \cos 2t\).
• Finding an expression for \(t\) in terms of one of the parametric variables. In this example, solving \(x = \sin t\) gives \(t = \arcsin x\).
• Substituting the expression back into the other parametric equation and using trigonometric identities to consolidate into one equation in terms of \(x\) and \(y\).

This process ensures that we understand the mathematical shape and position of the path in the \(xy\)-plane.

Understanding particle motion through parametric equations is key in visualizing how particles travel along their paths. These equations tell you where the particle is at any particular time \(t\). The motion is defined in two dimensions, \(x\) and \(y\), in terms of the parameter \(t\).
In this problem, as the parameter \(t\) changes from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the equations \(x = \sin t\) and \(y = \cos 2t\) describe the trajectory, specifying the horizontal and vertical coordinates at each moment.
The direction of motion is an important detail, which can be traced by checking how changes in \(t\) affect \(x\) and \(y\). We see that as \(t\) increases, \(x\) starts from \(-1\) and goes to \(1\). Observing \(y\), particularly its changes based on \(t\), helps to understand that the movement starts at one end of the parabola (\(-1\)), peaks, and then descends down the other side, confirming a left to right movement.

• Extremely useful in physics and engineering for modeling trajectories.
• Can easily account for changing velocities and directions with simple adjustments to parameters.

This understanding bridges theoretical equations and real-world path tracing.

The graph of a parabola is a visual representation of its equation. For the equation \(y = 1 - 2x^2\), the curve is quadratic and symmetric about the y-axis, opening downwards due to the negative coefficient of \(x^2\).
In the context of the parametric problem, graphing the parabola helps us visualize the path traced by the particle. The vertex of this parabola is at the point \((0, 1)\), which represents the highest point the particle reaches along its journey.
Graphing reveals several characteristics to keep in mind:

• The range of \(x\) from \(-1\) to \(1\) outlines the domain over which the particle travels.
• Because the parabola is symmetrical, half the graph shows movement from one side to the vertex, and the other half shows movement away from the vertex.
• The direction can be indicated by an arrow along the path, starting from \(-1\) towards \(1\), reflecting left-to-right motion.

By visualizing this graph, one gains insight into the particle's dynamics—where it speeds up, slows down, or changes direction—making the abstract equations come to life.