Problem 24 - Essential Calculus

Question

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=e^{2 t}, \quad y=e^{t} $$

Step-by-Step Solution

Answer

The corresponding rectangular equation is \( x = y^2 \). The graph, a parabola, is oriented to the right.

Step 1: Understand the Parametric Equations

The parametric equations given are \( x = e^{2t} \) and \( y = e^t \). In Cartesian coordinate system, each point is represented by two coordinates (x, y). In the parametric form, x and y are both expressed in terms of a third variable, often time 't'. Here, both x and y are functions of 't'.

Step 2: Eliminate the Parameter

To convert the parametric equations into rectangular form, the parameter needs to be eliminated. To do so, observe that the equation for \( x \) is simply the square of the equation for \( y \) (since squaring \( e^t \) gives \( e^{2t} \)). This lets us write \( x \) as a function of \( y \), giving us the rectangular equation \( x = y^2 \).

Step 3: Graph the Curve

Using a graphing utility, put in the equation \( x = y^2 \) to graph the curve. This will give a parabola opening to the right. The parametric equations indicate that as \( t \) increases, both \( x \) and \( y \) increase, indicating that the curve is oriented to the right.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps in visualizing mathematical equations and functions. When dealing with parametric equations, these utilities offer a simple way to translate the abstract equations into a comprehensible graph.
They are often available as software on computers or as apps on handheld devices and can beautifully render complex functions. Using a graphing utility to graph parametric equations involves inputting the equations and setting an appropriate range for the parameter, usually denoted as 't'. The graphing utility then plots points for various values of 't' and connects them, showing the curve's trajectory.
The orientation of the curve is crucial as it indicates the direction the curve is traced when the parameter increases. This orientation is easily observed by watching how the curve is drawn by the plotting tool.
These utilities often allow you to animate the plotting process, offering a real-time pause and play of how the curve develops, which can solidify one's understanding of the relationship between the parameter and the graph's shape.

Rectangular Equations

Rectangular equations, also known as Cartesian equations, describe a curve using the two variables x and y without a parameter like 't'.
In this scenario, the given parametric equations were converted into a rectangular equation by eliminating the parameter. This process is critical as it reduces the complexity of the problem, letting us work within the familiar Cartesian plane.The rectangular equation derived from the parametric equations was identified as \( x = y^2 \). This equation represents a parabola, a common curve in algebra. Unlike parabolas opening upwards or downwards seen in functions like \( y = x^2 \), this parabola opens to the right due to how x depends on y squared.
Understanding rectangular equations enables better comprehension of real-world phenomena as they offer a simpler way to analyze and interpret data and geometric shapes without always referencing a third variable.

Parameter Elimination

Parameter elimination refers to the process of removing the parameter from parametric equations to obtain a rectangular equation. This conversion is often necessary when one is required to interpret the graph in traditional x-y terms.
In our example, the parameter 't' was eliminated by recognizing the relationship between \( x \) and \( y \).With parametric equations \( x = e^{2t} \) and \( y = e^t \), the task involved expressing one variable in terms of the other. Since \( y = e^t \), by squaring y, we get \( y^2 = e^{2t} \). Then equating it to x as \( x = y^2 \), eliminates the parameter 't'.
This technique is a cornerstone in algebra and calculus, providing foundational skills crucial for further mathematical learning. It simplifies complex relationships involving time or other external variables, simplifying analysis and enabling you to recognize familiar geometric shapes like lines, circles, or parabolas in the Cartesian plane.