Problem 31 - Precalculus

Question

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=e^{-t}\\\ &y=e^{3 t} \end{aligned}$$

Step-by-Step Solution

Answer

The curve sketched by the parametric equations moves in the direction of increasing \( t \). By eliminating the parameter, we derived the rectangular equation \( y = x^{-3} \) for \( x > 0 \). This equation represents the same curve but in Cartesian form.

Step 1: Plotting the Parametric Equations

Plot the x and y-coordinates using the given parametric equations. When \( t = 0 \), \( x = e^0 = 1 \) and \( y = e^(3*0) = e^0 = 1 \), so plot the point (1,1). Also plot points for other values of \( t \) to get a feel of the curve. Mark the orientation towards increasing \( t \) to indicate the direction of the curve.

Step 2: Eliminating the Parameter

Eliminating the parameter requires relating \( x \) and \( y \) directly with each other. This can be done by solving one of the parametric equations for \( t \) and substitifying that into the other equation. Here, from \( x = e^{-t} \) we can derive that \( t = -ln(x) \). Replace \( t \) in the equation for \( y \) with this to get \( y = e^{3(-ln(x))} = e^{-3ln(x)} \), which simplifies to \( y = x^{-3} \). This is the rectangular equation.

Step 3: Adjusting the Domain of the Rectangular Equation

Looking at the plot, the curve does not cover negative x-coordinates as \( t \) ranges from negative infinity to positive infinity. Therefore, we restrict the domain to \( x > 0 \). The final rectangular equation is thus \( y = x^{-3} \), for \( x > 0 \). Note, this curve will approach the x-axis but never touch or cross it, reflecting the characteristic of the exponential function \( e^t \).

Key Concepts

Eliminating the Parameter

In parametric equations, each variable, such as \(x\) and \(y\), is expressed as a function of a parameter, in this case, \(t\). The goal of eliminating the parameter is to find a direct relationship between \(x\) and \(y\), without \(t\). This produces a rectangular equation, where \(x\) and \(y\) are related directly.

To eliminate the parameter in our example, begin with the equation for \(x\), which is \(x = e^{-t}\). Solve for \(t\) to get \(t = -\ln(x)\). Then, substitute \(t = -\ln(x)\) into the \(y\) equation, \(y = e^{3t}\). This transformation leads us to \(y = e^{3(-\ln(x))} = e^{-3\ln(x)}\). Simplifying further using logarithmic properties gives us \(y = x^{-3}\).

This process effectively removes \(t\) and provides a clean relationship between \(x\) and \(y\), making it simpler to understand their interaction.

Rectangular Equation

A rectangular equation directly relates \(x\) and \(y\) without the need for an additional parameter. It is a way to express the curve defined by parametric equations in the more familiar Cartesian coordinate system. In our solution, the rectangular equation derived is \(y = x^{-3}\).

This equation describes the curve initially given in parametric form. Here, each point on the curve satisfies this relation. Notice how the transformation from parametric to rectangular form has simplified the original description.

By eliminating \(t\) and reaching this equation, you can now analyze the curve using just the familiar \(x\) and \(y\) values, free of any parameters.

Domain of the Equation

When translating from parametric to rectangular form, it's essential to consider the domain of the new rectangular equation. The domain indicates the set of possible \(x\) values for which the function is defined.

In the case of the derived rectangular equation \(y = x^{-3}\), we must ensure that the domain aligns with the parametric origin. Initially, the parameter \(t\) could vary from negative to positive infinity. However, the equation \(x = e^{-t}\) yields only positive values for \(x\) because exponential functions are always positive. Thus, the domain of \(x\) in the rectangular equation is \(x > 0\).

This restriction on the domain reflects the characteristics of exponential functions—they never reach zero or negative values. As a result, when sketching or analyzing the rectangular equation, focus only on the positive \(x\)-axis direction.