Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=\ln 2 t, y=2 t^{2}$$

Step-by-Step Solution

Answer

The corresponding rectangular equation for the given parametric equations is \(y=2e^{x}\) for \(x\geq 0\).

Step 1: Graphing the parametric equations

Start by sketching a graph using the given parametric equations. You can generate points that fulfill the equation for several values of \(t)\). For instance, if \(t\) is 1, \(x=\ln(2)\) and \(y=2\). If you continue to plot points for a range of \(t\) values, you will get a graph that gives a clear indication of the curve’s orientation. Use a graphing utility software for accurate results.

Step 2: Converting to Rectangular Form

To convert the parametric form to rectangular form, first solve one of the given equations for \(t\) then substitute into the other equation. Let us solve the \(x\) equation for \(t\): \(t = e^{x/2}\), substituting \(t\) into the \(y\) equation, we get \(y = 2*(e^{x/2})^2 = 2e^x\).

Step 3: Adjusting the domain

This is necessary in some cases to properly represent the graph. For our current equation, the parametric form expresses that \(t \geq 0\). So, for the rectangular equation, since \(t = e^{x/2}\) and \(t \geq 0\), this means \(x/2 \geq 0\), or simply \(x \geq 0\).

Key Concepts

Rectangular Equation

Parametric equations are mathematical expressions where variables are expressed in terms of one or more parameters. In our given problem, the original parametric equations are \(x = \ln(2t)\) and \(y = 2t^2\). To transform these into a single rectangular equation, we eliminate the parameter \(t\).

The method involves solving one of the equations, say \(x = \ln(2t)\), for \(t\), which gives us \(t = e^{x/2}\). Substituting this into the second equation for \(y\), we find \(y = 2(e^{x/2})^2 = 2e^x\). This is our simplified rectangular equation that models the same curve as the parametric form.

This transformation to a rectangular equation is vital when exploring the properties of the curve without considering the parameter.

Graphing Utility

Sometimes, drawing curves manually using parametric equations can be cumbersome with the need to generate numerous data points. Here, a graphing utility can save the day. Graphing utilities, like GeoGebra or Desmos, allow users to visualize complex equations effortlessly by automating the calculation of points based on parameter values.

Using such a tool, we input the parametric equations \(x = \ln(2t)\) and \(y = 2t^2\) to generate a graphical representation of the curve. This visual confirmation can provide insights into the behavior and orientation of the curve and cross-verify manual computations.

These utilities are especially advantageous when handling a large dataset or when ensuring accuracy and efficiency in sketching graphs.

Graphing Parametric Equations

Graphing parametric equations involves plotting points by assigning different values to the parameter \(t\). For each value of \(t\), you calculate corresponding \(x\) and \(y\) values and plot these points on a coordinate plane.

In our exercise, begin with integer values of \(t\), such as \(t = 1\), to get \(x = \ln(2)\) and \(y = 2\). Continue this process for a range of \(t\) to map the entire curve. The resulting plot will often depict a smooth curve with an orientation, signifying the direction in which \(t\) increases.

This step-by-step graphical approach gives a better understanding of the parameter's influence on the curve's shape and direction, ensuring a comprehensive grasp of the geometric representation.

Eliminating the Parameter

Eliminating the parameter from parametric equations lets us convert them into a more traditional form, which is the rectangular equation. This approach primarily relies on algebraic manipulation to make one equation in terms of the other variable.

To tackle the parameter elimination, choose one equation and isolate the parameter \(t\). For example, with \(x = \ln(2t)\), solving for \(t\) yields \(t = e^{x/2}\). Then, substitute this expression into the other equation, \(y = 2t^2\), to derive our rectangular equivalent, \(y = 2e^x\).

This resultant equation is crucial for analyzing the curve in a non-parametric form and helps simplify complex parametric forms to enhance analyze-ability while maintaining the curve's essential characteristics.