Question

Use technology to sketch \(x=2 \tan (t), y=3 \sec (t),-\pi

Short answer

Graph the parametric equations using a software to see a hyperbola with vertical asymptotes at \(t = \frac{\pi}{2}\) and \(t = -\frac{\pi}{2}\).

Step-by-step solution

Understanding the Problem

We need to use a technology tool (such as a graphing calculator or software) to plot the parametric equations given: \(x = 2 \tan(t)\) and \(y = 3 \sec(t)\) over the interval \(-\pi < t < \pi\). This involves graphing a set of equations where \(x\) and \(y\) depend on a third parameter, \(t\).

Analyze the Functions

Identify how the functions work: \(\tan(t)\) and \(\sec(t)\) have vertical asymptotes where \(\cos(t) = 0\), which happens at \(t = \frac{\pi}{2}\) and \(t = -\frac{\pi}{2}\). These asymptotes partition the \(t\)-domain into separate intervals, which will reflect as undefined portions in the sketch.

Choosing Software

Select a graphing tool that allows parametric plotting. Tools like Desmos, GeoGebra, or a graphing calculator can be used. They generally have features to input parametric equations directly.

Input the Equations

Open the graphing tool and enter the parametric equations. Many tools ask for \((x(t), y(t))\), so input "x = 2 * tan(t)" and "y = 3 * sec(t)". Set the parameter \(t\) to vary within the interval \(-\pi < t < \pi\).

Analyze the Graph

Observe the plotted graph. The graph will show a hyperbolic shape with vertical asymptotes at \(t = \frac{\pi}{2}\) and \(t = -\frac{\pi}{2}\), where both \(\tan(t)\) and \(\sec(t)\) become undefined. The graph will be symmetric with respect to the x-axis due to the symmetry of \(\tan(t)\) and \(\sec(t)\).

Key concepts

Graphing Techniques

Graphing parametric equations involves plotting points based on an independent variable parameter, often denoted by \( t \). This differs from regular Cartesian graphs where you have a direct \( y = f(x) \) relationship. With parametric equations, each coordinate (\( x \) and \( y \)) is separately expressed as functions of \( t \). To graph such equations, you use a graphing calculator or software capable of parametric plotting.

When graphing, you first select the appropriate tool. Next, input the equations for \( x(t) \) and \( y(t) \). It's essential to define the range for \( t \), which in our case is from \(-\pi\) to \(\pi\). As the parameter \( t \) varies, the graphing tool calculates corresponding \( x \) and \( y \) values. The beauty of tech-aided graphing is its ability to handle complex equations, effectively showing behaviors like asymptotes and symmetry, which might be tedious to calculate by hand.

Trigonometric Functions

Trigonometric functions like \( \tan(t) \) and \( \sec(t) \) play a crucial role in parametric equations. The tangent function, \( \tan(t) \), is defined as the ratio of sine to cosine, \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). It is periodic with a period of \( \pi \), which causes it to repeat its behavior every \( \pi \) units.

The secant function, \( \sec(t) \), is the reciprocal of the cosine function, \( \sec(t) = \frac{1}{\cos(t)} \). It inherits the properties of cosine, particularly undefined points where \( \cos(t) = 0 \). Both functions display vertical asymptotes at these points. For our exercise, these are at \( t = \frac{\pi}{2} \) and \( t = -\frac{\pi}{2} \).

Since \( \tan(t) \) has an odd symmetry and \( \sec(t) \) has even symmetry, when you graph them together in parametric form, the resulting curve often has interesting symmetric properties. Knowing these fundamental trigonometric identities helps in predicting and understanding the plotted shapes.

Asymptotes

Asymptotes are lines that a curve approaches but never actually touches or crosses. In the graph of parametric equations containing \( \tan(t) \) and \( \sec(t) \), vertical asymptotes occur where these functions are undefined.

For \( \tan(t) \) and \( \sec(t) \), vertical asymptotes appear at \( t = \frac{\pi}{2} \) and \( t = -\frac{\pi}{2} \) because \( \cos(t) = 0 \) at these points. This results in undefined values—\( \tan(t) \) tends towards positive or negative infinity, and \( \sec(t) \) has discontinuities, mirrored in the sketched graph as lines that \( x \) and \( y \) approach but do not touch.

Understanding these asymptotes is key in predicting the appearance of the graph, avoiding any misleading assumptions about the behavior of the curve beyond these undefined points. Asymptotic behavior often dictates the structure of the graph and its segments, ensuring that calculations remain anchored in the realms of real, defined values.