Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Graph each pair of parametric equations for \(0 \leq t \leq 2 \pi\) in the window \([0,6.6]\) by \([0,4.1] .\) Identify the letter of the alphabet that is being graphed. $$\begin{aligned} &x_{1}=1, \quad y_{1}=1+\frac{t}{\pi}\\\ &x_{2}=1+\frac{t}{3 \pi}, \quad y_{2}=2\\\ &x_{3}=1+\frac{t}{2 \pi}, \quad y_{3}=3 \end{aligned}$$

Step-by-Step Solution

Answer

The graph forms the capital letter 'T'.

Step 1: Parameter Range Setup

The given parametric equations need to be evaluated over the range of \(0 \leq t \leq 2\pi\). This defines our domain for plotting each curve.

Step 2: Plot the First Equation

For the first equation, we have \(x_1 = 1\) and \(y_1 = 1 + \frac{t}{\pi}\). This represents a vertical line because \(x_1\) is a constant value (1) while \(y_1\) changes linearly with \(t\). As \(t\) increases from 0 to \(2\pi\), \(y_1\) will increase from 1 to 3.

Step 3: Plot the Second Equation

For the second pair of equations, we have \(x_2 = 1 + \frac{t}{3\pi}\) and \(y_2 = 2\). This equation yields a horizontal line since \(y_2\) is constant while \(x_2\) changes linearly with \(t\). As \(t\) increases from 0 to \(2\pi\), \(x_2\) will range from 1 to approximately 2.67.

Step 4: Plot the Third Equation

In the third set, \(x_3 = 1 + \frac{t}{2\pi}\) and \(y_3 = 3\). Similarly, this forms another horizontal line with \(y_3\) as a constant. Here, as \(t\) increases from 0 to \(2\pi\), \(x_3\) will vary from 1 to 2.

Step 5: Identify the Letter

By combining all these plots: a vertical line at \(x = 1\) spanning from \(y = 1\) to \(y = 3\), a horizontal line at \(y = 2\) from \(x = 1\) to approximately 2.67, and another horizontal line at \(y = 3\) from \(x = 1\) to 2. This creates the shape of the capital letter \(T\).

Key Concepts

Graphing

Graphing parametric equations is a wonderful way to visualize relationships between two varied functions over a particular range. When graphing the equations

• \(x_1 = 1, \, y_1 = 1 + \frac{t}{\pi}\)
• \(x_2 = 1 + \frac{t}{3\pi}, \, y_2 = 2\)
• \(x_3 = 1 + \frac{t}{2\pi}, \, y_3 = 3\)

we approach this by considering how each pair of equations behaves as the parameter\(t\) varies from 0 to \(2\pi\).
In this context, we are dealing with movements confined to the Cartesian plane. It's insightful to note that each equation can be understood as a distinct line on the graph. For instance, the first equation plots a vertical line because the \(x_1\) coordinate remains constant while \(y_1\) changes. The second and third equations plot horizontal lines where the \(y\) coordinate stays constant as \(x_2\) and \(x_3\) change. Graphing brings these relationships to life, enabling you to see the geometric representation of algebraic equations. It becomes even more fascinating when you realize these lines come together to form specific shapes, like letters!

Coordinate System

The coordinate system is pivotal when working with parametric equations, especially when converting the intrinsic relationship between two variables into a visual form. Here, we're discussing a 2D system with x and y axes.
The interplay of components such as constants and variables in the equations

• \(x = 1, \, y = 1 + \frac{t}{\pi}\)
• \(x = 1 + \frac{t}{3\pi}, \, y = 2\)
• \(x = 1 + \frac{t}{2\pi}, \, y = 3\)

allows movements along both the horizontal and vertical axes. Each increment in \(t\) corresponds to a change in either the x or y values or both, depending on the equation.
This manipulation of variables demonstrates that the Cartesian coordinate system isn't just about placing dots on a grid. It's a versatile framework allowing us to describe positions and draw meaningful pictures just like drafting alphabets which rely on specific configurations!

Alphabet Letter Identification

Identifying an alphabet letter from graphs of parametric equations is an interesting task, quite like solving a geometric puzzle! For these equations

• A vertical line at \(x = 1\), from \(y = 1\) to \(y = 3\)
• A horizontal line at \(y = 2\), spanning from \(x = 1\) to approximately \(x = 2.67\)
• Another horizontal line at \(y = 3\), from \(x = 1\) to \(x = 2\)

this particular configuration resembles the capital letter 'T'. These distinct lines—one vertical and two horizontal—come together to visually form the classic 'T' shape that is familiar in the alphabet.
This ability to visually connect mathematical equations to recognizable shapes like letters enhances our understanding of the unity between algebra and geometry, enriching the perceptive and analytical skills we use in mathematics.