Problem 5 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(5-22,\) a parametrization is given for a curve. (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? $$x=3 t, \quad y=9 t^{2}, \quad-\infty

Step-by-Step Solution

Answer

The parametric equations \(x = 3t\) and \(y = 9t^2\) represents the entire graph of the Cartesian equation \(y = x^2\), with no initial and terminal points.

Step 1: Sketch the Parametric Plot and Find Initial and Terminal Points

The given parametric equations are \(x = 3t\) and \(y = 9t^2\). As the domain of \(t\) extends from \(-\infty\) to \(+\infty\), the initial and terminal points don't exist. The graph can be plotted by substituting various values of \(t\) into \(x\) and \(y\). The direction of the curve is determined by the increasing values of \(t\).

Step 2: Convert the Parametric Equation to Cartesian Form

To find the Cartesian form, one can first solve one of the parametric equations for \(t\) and then substitute that expression into the other parametric equation. Solving the \(x\)-equation for \(t\) gives \(t = x/3\). Substituting \(t = x/3\) into the \(y\)-equation, gives \(y = 9(x/3)^2 = x^2\).

Step 3: Determine the Traced Portion of the Cartesian Graph

As the domain of \(t\) is \(-\infty < t < +\infty\), the graph of the Cartesian equation \(y = x^2\) is entirely traced by the parametric curve.

Key Concepts

Parametrization of Curves

Understanding the concept of parametrization is essential for grappling with the problems of graphing curves. Parametrization means expressing the coordinates of the points on a curve as functions of a single parameter, often denoted as \( t \). This parameter traces the path along the curve, which can represent time or another continuous variable.

In our exercise, the curve is defined using parametric equations \( x = 3t \) and \( y = 9t^2 \). By changing the value of \( t \), we can determine different points along the curve. This technique is powerful in describing curves that might not be easily described by simple Cartesian equations.

• **Advantages of Parametrization:** It allows more flexibility and control over tracing a curve, specifying direction, and sometimes even the speed.
• **Application:** Useful when dealing with curves like ellipses and parabolas, where standard forms might not be sufficient.

As \( t \) varies from \(-\infty\) to \(+\infty\), the curve described by these particular parametric equations changes position in a specific direction, capturing all integral parts of the parabola described.

Cartesian Equation

The Cartesian equation of a curve provides a relationship between \( x \) and \( y \) in the familiar form used in standard graphing. It is derived from the parametric equations by eliminating the parameter \( t \).

For the given exercise, the transformation from parametric to Cartesian involves solving one parametric equation for \( t \). From \( x = 3t \), we find \( t = x/3 \). By substituting \( t = x/3 \) into \( y = 9t^2 \), we get \( y = 9(x/3)^2 = x^2 \).

• **Importance in Graphing:** Provides a single function representation of the curve, making it easier to understand relationships and plot using different methods or tools.
• **Visual Clarity:** Most graphing tools and our visual intuition are better tuned to interpret Cartesian equations, making \( y = x^2 \) particularly useful for sketching.

Every point defined by the parametric equations fits this Cartesian equation, showing that they represent the same curve.

Graph Sketching

Graph sketching is the process of drawing the curve defined by the equations, giving a visual interpretation of the parametric or Cartesian equations.

When sketching curves for this specific problem, you begin by considering several values for \( t \), calculate the corresponding \( x \) and \( y \) coordinates, and then plot these points. As \( t \) increases, the path of the curve continues, giving insight into its shape and direction.

• **Directionality:** In the exercise, the curve direction is indicated as the direction \( t \) increases, which goes from left to right since \( x = 3t \) grows.
• **Smooth Transition:** Parametric equations often suggest a smooth path or transition on curves like parabolas, making them intuitive to follow.

The plotted curve in this case is a parabola opening upwards since \( y = x^2 \), traced completely as described by the parametric form.

Initial and Terminal Points

Initial and terminal points refer to the starting and ending location on a curve as defined by a range of parameter \( t \).

In some cases, these might be clearly defined, but when \( t \) ranges from \(-\infty\) to \(+\infty\), as in this exercise, such specific points do not exist. The curve continues indefinitely in both directions on the \( x \)-axis.

• **No Fixed Bounds:** Without a set endpoint for \( t \), the curve is infinite, illustrating the difference between bounded and unbounded parametric curves.
• **Continuous Pathway:** For graphing purposes, understanding this can be essential in recognizing curves that do not "end" but continue to reoccur through further ranges of \( t \).

Overall, recognizing the continuum of the curve rather than fixed initial and terminal points can aid in understanding the full extension and implications of the graph.