Problem 59 - Precalculus: Functions and Graphs

Question

Graph the curve. $$x=3 t-2 \sin t, \quad y=3-2 \cos t, \quad-8 \leq t \leq 8$$

Step-by-Step Solution

Answer

Graph the curve using parametric equations for values of \(t\) from -8 to 8.

Step 1: Understand the Parametric Equations

We are given parametric equations for a curve where the values of \(x\) and \(y\) depend on the parameter \(t\). Specifically, \(x = 3t - 2\sin(t)\) and \(y = 3 - 2\cos(t)\). Our goal is to plot these equations over the domain \(-8 \leq t \leq 8\).

Step 2: Select Values for t

To graph the curve, we select several values for \(t\) between -8 and 8. This might include whole numbers, and some specific fractional or decimal values that are easy to compute.

Step 3: Calculate x and y for Chosen t Values

For each selected \(t\), compute the corresponding \(x\) and \(y\) coordinates using the parametric equations: \(x = 3t - 2\sin(t)\) and \(y = 3 - 2\cos(t)\).

Step 4: Plot the Points

Using the calculated \(x\) and \(y\) values, plot the points on the Cartesian coordinate plane. This gives a visual representation of the curve.

Step 5: Connect the Points

Connect the plotted points smoothly to form the curve. Since trigonometric functions are involved, expect periodic behavior.

Key Concepts

Graphing Techniques

Graphing techniques are a valuable skill when dealing with parametric equations. These equations express coordinates \(x\) and \(y\) as functions of a parameter \(t\). To graph a parametric curve, you start by selecting a range of \(t\) values where the parameter varies. Here, we consider \(-8 \leq t \leq 8\).

• Choose a variety of \(t\) values: include negative, zero, positive, and fractional numbers if needed, to capture the curve's behavior at every segment.
• Compute corresponding \(x\) and \(y\) coordinates by substituting \(t\) values into the parametric equations.
• Once the coordinates are determined, plot each point onto a Cartesian coordinate system.
• Connect the dots smoothly, since parametric equations often represent continuous curves. Be mindful of the periodic nature, such as when trigonometric functions are involved, which creates waves or loops.

Graphing requires patience and precision, ensuring each step is carefully followed to accurately reflect the curve defined by the parametric equations.

Trigonometric Functions

Trigonometric functions like sine and cosine are powerful tools in parametric equations. These functions handle periodic changes, adding dynamic patterns to the curve's shape. In our exercise, the equations are structured as

• \(x = 3t - 2\sin(t)\), which translates shifts and oscillations horizontally.
• \(y = 3 - 2\cos(t)\), manages the vertical movement through its maximums (at amplitude peak of cosine) and minimums.

These functions help in modeling real-world phenomena involving cycles and rotations, such as waves or circular motions. Understanding where sine and cosine reach their extremes helps anticipate the peaks and troughs of the plotted points. Key points about sine and cosine:

• Sine of an angle achieves maximum of \(1\) and minimum of \(-1\).
• Cosine, similar in amplitude, follows the same range from \(1\) down to \(-1\).
• They both complete a full cycle over an interval of \(2\pi\).

Incorporating these insights into graphing processes ensures an appreciation of the trigonometric influence on the overall curve.

Cartesian Coordinate System

The Cartesian coordinate system is the canvas where parametric equations come alive through graphing. This system involves two axes: the horizontal \(x\)-axis and the vertical \(y\)-axis, intersecting at the origin \( (0, 0) \). In plotting our parametric equations, you rely on this grid to display points you've calculated. Steps for using the Cartesian coordinate system effectively include:

• Understanding the scale: Adjust the axis units as needed for clarity and full view of the curve.
• Plotting accuracy: Mark each computed point accurately where its \(x\) value and \(y\) value intersect.
• Utilize the quadrants: By knowing which quadrant the curve segments extend into, you foresee the behavior and flow of the line formed.
• Smooth connections: Draw lines or curves connecting the points in accordance with the expected behavior dictated by the parametric form.

This system underpins the art of graphing, transforming a set of abstract numbers and equations into a visible and meaningful curve. The Cartesian plane provides clarity and context, showcasing the beauty and intricacy of parametric curves.