Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=t, y=\frac{1}{t},\) for \(t\) in \([-10,10]\)window: \([-6,6]\) by \([-4,4]\)

Step-by-Step Solution

Answer

Rectangular equation: \(y = \frac{1}{x}\).

Step 1: Understanding Parametric Equations

The given parametric equations are \(x = t\) and \(y = \frac{1}{t}\). This means that as \(t\) varies, \(x\) and \(y\) get their coordinates based on \(t\)'s value. Our task is to understand each of these variables in terms of \(t\) and their relationship.

Step 2: Graph the Parametric Equations

Use your graphing calculator to input the parametric equations \(x = t\) and \(y = \frac{1}{t}\). Make sure your calculator window is set to the specified limits: \([-6,6]\) for \(x\) and \([-4,4]\) for \(y\). Observe the shape of the graph as \(t\) ranges from -10 to 10.

Step 3: Convert Parametric Equations to Rectangular Form

Since \(x = t\), substitute \(t\) in the equation for \(y\). This gives \(y = \frac{1}{x}\). This is the rectangular equation for the curve.

Step 4: Verify the Graph

With the rectangular equation \(y = \frac{1}{x}\), use the graphing calculator to plot this equation over the same window. Verify that the shape matches the curve generated by the parametric equations over the interval \(t\).

Key Concepts

Rectangular Equation

A rectangular equation is essentially an equation that relates two variables, typically denoted as \(x\) and \(y\), in the coordinate plane. In this exercise, we start with parametric equations: \(x = t\) and \(y = \frac{1}{t}\). Our goal is to eliminate the parameter \(t\) and express \(y\) solely in terms of \(x\). By recognizing that \(x = t\), we simply substitute \(x\) for \(t\) in the equation for \(y\), resulting in \(y = \frac{1}{x}\).
This transformation shifts our perspective from a dependence on the parameter \(t\) to focusing directly on the standard Cartesian coordinates \(x\) and \(y\). This rectangular equation is now expressed in a form that can be readily visualized or further manipulated in the context of a coordinate plane. Understanding and converting parametric equations to rectangular form is a powerful technique, as it provides a unified method to analyze and interpret different types of curves.

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical concepts like parametric equations and rectangular equations. By inputting the given parametric equations \(x = t\) and \(y = \frac{1}{t}\) into the calculator, students can observe the resulting graph. It's critical to set the graphing window correctly to see the full picture. In this case, use a window with \(x\) limits of \([-6, 6]\) and \(y\) limits of \([-4, 4]\).
Graphing calculators allow for dynamic exploration:

• You can adjust the limits to see how the graph changes.
• You can explore different values of \(t\) to observe their effect on \(x\) and \(y\)

These functionalities make graphing calculators invaluable for solving complex mathematical problems and understanding their graphical representations. After plotting the parametric equations, you can also graph the rectangular equation \(y = \frac{1}{x}\) to see if it matches your previous graph over the given interval.

Graph Interpretation

Interpreting graphs involves analyzing the visual representation created when equations are plotted on the coordinate plane. When you graph the parametric equations \(x = t\) and \(y = \frac{1}{t}\), you'll notice the graph is a hyperbola, reflecting the inverse relationship described by \(y = \frac{1}{x}\). This appears as two distinct branches, one in the first and one in the third quadrants.
Important aspects of graph interpretation:

• Identify symmetries – here, the graph is symmetric with respect to the origin.
• Understand the limits of the graph as \(t\) approaches extreme values, like \(-10\) and \(10\), where \(y\) becomes very large in magnitude.

Recognizing these features helps in grasping how equations translate into shapes. It reveals the underlying relationships between variables, such as how the graph of \(y = \frac{1}{x}\) doesn't touch the axes, but approaches them asymptotically. Appreciating these subtleties is vital for a thorough understanding of calculus and algebra.