Question

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. Be sure to specify the interval over which the parameter varies. The complete curve \(x=y^{3}-3 y\)

Short answer

Short Answer: The curve \(x=y^3-3y\) can be represented by the parametric equations \(x = t^3 - 3t\) and \(y = t\), where the positive orientation is upwards. The parameter t varies over the entire real number range \((-\infty, \infty)\). The graph of the curve has a "W" shape.

Step-by-step solution

Parametric Representation of the Curve

A parametric representation of a curve involves rewriting the given equation in terms of a parameter (usually denoted by t). In this case, we can choose \(y=t\). Therefore, the equation for x becomes \(x=t^3-3t\). Thus, the parametric equations that describe the curve are: $$x = t^3 - 3t$$ $$y = t$$

Determine Positive Orientation

The positive orientation of a curve is determined by observing how the curve is traversed as the parameter (in this case, t) increases. To see this, let's examine the curve at two particular points, \(t_1\) and \(t_2\) such that \(t_1 < t_2\). For \(t_1\), we have \((x_1,y_1) = (t_1^3-3t_1,t_1)\). For \(t_2\), we have \((x_2,y_2) = (t_2^3-3t_2,t_2)\). When \(t_1<t_2\), we have \(x_1<t_1^3-3t_1 < t_2^3-3t_2\), so the curve moves from left to right along the x-axis (positive half) as the parameter t increases. Thus, the positive orientation indicates upward movement in this case.

Specifying the Parameter Interval

In this case, we want to cover the entire curve as the parameter varies. Since the initial equation involves a cubic function which includes all values of x, we should specify that t varies over all real numbers. Therefore, the interval for t is: $$ t \in (-\infty, \infty) $$

Graph the Curve

Graph the complete curve \(x=y^3-3y\) with the given parametric equations and the specified interval for t. Follow the curve from left to right (the positive half) while increasing t, and observe the upward orientation. The graph will have a "W" shape. In conclusion, the parametric equations that describe the complete curve \(x=y^3-3y\) are: $$x = t^3 - 3t$$ $$y = t,$$ with a positive orientation pointing upwards and the parameter t varies over the interval \((-\infty, \infty)\).

Key concepts

Positive Orientation

When graphing parametric equations, it's important to understand the concept of orientation. Positive orientation refers to the direction in which a curve is traversed as the parameter increases. In this case, our parameter is denoted by \( t \). As \( t \) increases, the path followed by the curve can either move in a clockwise or counterclockwise direction for closed curves, or simply from left to right or upward along the axes for open curves.
To determine positive orientation, let's consider two values of \( t \), namely \( t_1 \) and \( t_2 \), where \( t_1 < t_2 \). This exercise reveals that as \( t \) increases, for \( t_1 < t_2 \), the corresponding \( x \, \) coordinates \( t_1^3 - 3t_1 \) and \( t_2^3 - 3t_2 \) equate to \( x_1 < x_2 \). This means the curve moves from left to right.
Ultimately, understanding positive orientation is crucial for determining how to traverse a parametric curve correctly as \( t \) changes.

Parameter Interval

The parameter interval is the range of values that the parameter \( t \) can take. It defines the part of the graph that the parametric equations will cover. By focusing on this interval, we ensure that the entire part of the curve we are interested in is shown, and no points are excluded.
In this example, the curve equation is \( x = y^3 - 3y \), which implies a cubic function. Cubic functions typically extend from \(-\infty \) to \(\infty\), presenting all real numbers as possible outcomes. Therefore, the parameter \( t \) should also vary across all real numbers to capture the complete behavior of the curve. Thus, \( t \in (-\infty, \infty) \).
Grasping the importance of the parameter interval in parametric equations allows for a complete understanding of the curve's behavior.

Graphing Parametric Curves

Graphing parametric curves involves plotting points on a coordinate system where each point is determined by parametric equations. The values of \( x \) and \( y \) depend on \( t \), and as \( t \) changes, so does the position of these points, forming a curve.
For our curve, described with the equations \( x = t^3 - 3t \) and \( y = t \), the graph will have a signature "W" shape. Begin plotting the curve by calculating a handful of points, choosing various values of \( t \) both positive and negative, and observing how \( x \) and \( y \) change. This step is vital to determine how the curve traverses through the graph.
Remember, when graphing, always pay attention to the positive orientation and parameter interval. These ensure you are correctly depicting how the curve moves and determine its complete shape over the desired range. With these tools, interpreting and sketching parametric curves becomes intuitive and effective.