91Ó°ÊÓ

To grasp the concept of Cartesian Form, it's essential to understand how it helps connect parametric equations, which involve a parameter like \(t\), into equations containing only \(x\) and \(y\). In this exercise, we start with the parametric equations \(x = t^3\) and \(y = t^2\). To convert these into a Cartesian form, substitute the parameter \(t\) with an expression relating \(x\) and \(y\) directly. From the equation \(y = t^2\), we can solve for \(t\) by taking the square root, i.e., \(t = \pm \sqrt{y}\). Substituting these into \(x = t^3\), we obtain two expressions:

• For \(t = \sqrt{y}\), \(x = y^{3/2}\)
• For \(t = -\sqrt{y}\), \(x = -y^{3/2}\)

The Cartesian forms \(x = y^{3/2}\) and \(x = -y^{3/2}\) provide a complete description of the curve \(C\) in the \(xy\)-plane, representing the relationship between \(x\) and \(y\) without the need for \(t\). This conversion is crucial for visualizing the curve and analyzing its properties.

Understanding curve orientation is pivotal as it indicates the direction in which the curve is traced as the parameter changes. For the equations \(x = t^3\) and \(y = t^2\), the curve's orientation is determined by how \(x\) and \(y\) progress with differing values of \(t\). In our case, as \(t\) increases:

• \(x = t^3\) increases, reflecting movement towards the positive \(x\)-axis.
• Since \(y = t^2\) is always positive or zero, \(t\) effectively describes how the curve shifts along the \(y\)-axis as well.

The curve begins from the left, tracing from a negative \(x\) towards zero where \(t = 0\), then moves to positive \(x\) values as \(t\) continues to increase positively. On the curve, this looks like a sweep from \((-\infty, 0]\) to \([0, \infty)\) on the \(y\)-axis, making the orientation from left to right, much like following an 'S' shape on its journey across the plane.

Sketching a graph informs us visually about the behavior and structure of equations expressed in both parametric and Cartesian forms. The given curve is a classic example known as the Tschirnhausen cubic or 'semicubic parabola.' The graph of \(x = y^{3/2}\) and \(x = -y^{3/2}\) visually appears like a horizontally stretched 'S' across the \(y\)-axis. Here's how you can sketch it:

• Start by plotting basic points, focusing on the orientation established previously.
• Keep in mind that for both the positive and negative square root functions, the curve remains symmetric relative to the \(y\)-axis.
• The shape opens sideways, growing wider as \(y\) increases.

The orientation guides you to trace the curve effectively: begin from the negative side of the \(x\)-axis and steadily move towards the positive side, creating a mirrored path. The symmetry and orientation together depict the breadth of the curve, emphasizing its unique cubic nature. Keeping these steps and elements in mind empowers you to sketch and understand similar curves with confidence.