Question

Use technology to graph the curve given by the parametric equations \(x=2 \cot (t), y=1-\cos (2 t),-\pi / 2 \leq t \leq \pi / 2\). This curve is known as the witch of Agnesi.

Short answer

Graph the parametric equations using a graphing tool, ensuring symmetry and handling vertical asymptotes.

Step-by-step solution

Understand the Parametric Equations

The curve is defined by the parametric equations \(x = 2 \cot(t)\) and \(y = 1 - \cos(2t)\). Our task is to graph this curve over the domain \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\). Here, the parameter \(t\) represents the angle in trigonometric functions that will affect both \(x\) and \(y\) values.

Graphing Tools Setup

To graph the curve, you can use graphing software or a graphing calculator. Importantly, ensure that the graphing tool can handle parametric equations by setting \(x(t) = 2 \cot(t)\) and \(y(t) = 1 - \cos(2t)\), following the range of \(t\) from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Configure Plotting Range and Scale

As \( \cot(t) \) has vertical asymptotes at \( t = 0, \pm \frac{\pi}{2} \), you should handle these in the graphing tool settings. Also, adjust your plot range and scales appropriately to visualize the behavior near these points. Generally, \(x\) will run between \(-2 \sin(\pm \frac{\pi}{2})\) and close to infinity.

Execute Graphing

Run the graphing tool to generate the plot of the given parametric equations. Check to see the symmetry of the curve about the y-axis, as should be expected from the nature of trigonometric functions used in the equation.

Analyze the Curve Shape

The witch of Agnesi should appear as a smooth curve that loops around the y-axis, illustrating a single curve reaching from positive to negative infinity in the x-axis, as predicted by \( \cot(t) \). The curve's peak should be centered on the y-axis.

Key concepts

Trigonometric Functions

Trigonometric functions are at the heart of this exercise. They are mathematical functions that relate angles to the ratios of the sides of a triangle. In our parametric equations, we specifically use the cotangent and cosine functions. The cotangent function, \(\cot(t)\), is the reciprocal of the tangent function and is defined as \( \cot(t) = \frac{1}{\tan(t)} = \frac{\cos(t)}{\sin(t)} \). It has distinct characteristics such as vertical asymptotes at \( t = n\pi \), where \( n \) is any integer.
\(\cos(2t)\) is another trigonometric function we encounter in the equation, relevant because it modifies the typical cosine wave to shrink its period. Remember, \( \cos(2t)\) completes two full cycles in the same period where \(\cos(t)\) would complete only one.
To understand how these functions impact the shape of the witch of Agnesi, visualize how \(\cot(t)\) stretches out the x-values to infinity when facing its vertical asymptotes. Similarly, \(1 - \cos(2t)\) shifts the classic cosine function upwards by 1, impacting the y-value range. Hence, each value of \(t\) affects both the \(x\) and \(y\) coordinates, creating the unique curve.

Graphing Technology

Graphing technology is incredibly helpful for visualizing complex mathematical equations such as parametric equations. Whether using a graphing calculator or graphing software, these tools allow us to conveniently input equations and observe their graphical interpretations instantly. They work by evaluating each function at numerous points within a specified domain.
To graph a parametric equation like \(x = 2 \cot(t)\) and \(y = 1 - \cos(2t)\), you must configure the tool to accept parametric inputs. Ensure the software can handle the domain boundary conditions, here between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), and that it can manage vertical asymptotes by correctly adjusting the plotting scales.

• Choose a reliable graphing tool that supports trigonometric functions and parametric equations.
• Input the \(x(t)\) and \(y(t)\) definitions appropriately.
• Adjust scales to accurately capture behavior around asymptotes.

Once set up correctly, you should instantly see how the witch of Agnesi is realized graphically, aiding both your understanding and analysis.

Witch of Agnesi

The witch of Agnesi is a famous curve in mathematics, named after Maria Gaetana Agnesi. Although there are several stories about its misinterpretation leading to the term "witch," the focus is on its elegant properties and applications.
This curve is a smooth, bell-shaped figure notable in the study of calculus for illustrating principles of tangent lines and areas beneath curves. In our case, its parametric form is explored using trigonometric components. The usage of \(\cot(t)\) contributes to the infinite horizontal spread, while the modified cosine, \(1-\cos(2t)\), creates the distinctive peak and symmetry about the y-axis.
The witch of Agnesi is often used as an exercise to practice parametric graphing and to highlight important mathematical concepts such as symmetry, asymptotic behavior, and periodicity. While it appears complex, breaking it down into its components simplifies its understanding and enhances visualization skills in advanced math contexts.