Problem 44 - 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. $$\begin{aligned} &x_{1}=2+0.8 \cos 0.85 t, \quad y_{1}=2+\sin 0.85 t\\\ &x_{2}=1.2+\frac{t}{1.3 \pi}, \quad y_{2}=2 \end{aligned}$$

Step-by-Step Solution

Answer

The letter 'D' is graphed.

Step 1: Understanding the Parametric Equations

We are given two pairs of equations. The first pair is \(x_1 = 2 + 0.8 \cos(0.85t)\) and \(y_1 = 2 + \sin(0.85t)\). These equations describe a curve. The second pair is \(x_2 = 1.2 + \frac{t}{1.3 \pi}\) and \(y_2 = 2\). This describes a horizontal line at \(y=2\).

Step 2: Graphing First Parametric Equation

For the first pair, as \(t\) varies from \(0\) to \(2\pi\), the equations describe an ellipse centered at \((2,2)\) with a horizontal semi-axis length of \(0.8\) and a vertical semi-axis length of \(1\). Plot points as \(t\) progresses using these increments to map out the curve within the specified range.

Step 3: Graphing Second Parametric Equation

The second pair describes a straight line segment. As \(t\) increases from \(0\) to \(2 \pi\), \(x_2\) increases linearly starting at \(x=1.2\). This creates a horizontal line segment moving to the right, maintaining \(y=2\) throughout, spanning from \((1.2, 2)\) to approximately \((2.986, 2)\).

Step 4: Identifying the Alphabet Letter

By examining and combining the two shapes of the parametric curves, the resulting figure resembles the letter 'D'. The elliptical portion forms the curved part and the line segment forms the straight portion of 'D'.

Key Concepts

Ellipse

An ellipse is a geometric shape that looks like a stretched circle. It has two axes: a major axis and a minor axis. The major axis is the longest diameter, running through the center and touching the ellipse from one side to the other. The minor axis is the shortest diameter, perpendicular to the major axis.

When talking about parametric equations for an ellipse, like in the equations \(x_1 = 2 + 0.8 \cos(0.85t)\) and \(y_1 = 2 + \sin(0.85t)\), we see that the ellipse's center is at the point \((2, 2)\). The coefficients of \(\cos(0.85t)\) and \(\sin(0.85t)\) help determine the shape of the ellipse:

• The horizontal semi-axis length is 0.8 due to the coefficient 0.8 in front of \(\cos\).
• The vertical semi-axis length is 1 because the coefficient in front of \(\sin\) is 1.

This ellipse is horizontal, meaning the longer axis is along the x-direction, keeping the y-direction slightly shorter.
Understanding ellipses and their orientation is crucial for graphing and interpreting parametric curves correctly.

Graphing

Graphing parametric equations involves plotting points derived from equations over a set interval for a parameter, often taken as \(t\), which can stand for time. This process can turn a mathematical equation into a visual representation.

To graph the ellipse given by \(x_1 = 2 + 0.8 \cos(0.85t)\) and \(y_1 = 2 + \sin(0.85t)\), you carry out the following steps:

• Calculate the coordinates for a series of values of \(t\) ranging from 0 to \(2\pi\).
• Plot these points on a graph, marking the x values from the first equation and the y values from the second equation.
• Connect these points smoothly, considering the parametric form suggests an elliptical motion.

The result is an ellipse centered at \((2, 2)\), giving us a visual picture based on theoretical calculations.

Graphing simplifies complex parametric equations into intuitive shapes and helps in identifying patterns or characters these shapes might form.

Horizontal Line

A horizontal line is a straight line where all points have the same y-coordinate. It's easily recognized for its constant horizontal movement.

In this case, the second pair of parametric equations given by \(x_2 = 1.2 + \frac{t}{1.3 \pi}\) and \(y_2 = 2\) describes a horizontal line. Here’s how it is formed:

• As \(t\) increases from 0 to \(2\pi\), \(x_2\) grows linearly, starting at 1.2. This linear term, \(\frac{t}{1.3 \pi}\), ensures as time \(t\) progresses, \(x\) moves along the x-axis.
• The y-value remains constant at 2, demonstrating it as a horizontal line parallel to the x-axis.

Such lines are simple yet significant, as they often serve as boundaries, connectors, or parts of larger graphical figures. In parametric plots, they can complement more complex curves as seen by how this line combines with the ellipse to create the letter 'D'.