Problem 7 - Contemporary Precalculus

Question

Find a viewing window that shows a complete graph of the curve. $$x=\cos 3 t+t, \quad y=\cos 4 t+t, \quad-\pi \leq t \leq \pi$$

Step-by-Step Solution

Answer

Answer: The viewing window for the graph of the given parametric curve should be x: [-π-1, π+1] and y: [-π-1, π+1].

Step 1: Analyze the x equation

Since x = cos(3t) + t, we should find the minimum and maximum values of x for the given range of t (-π ≤ t ≤ π). The cos(3t) term ranges from -1 to 1. Therefore, the minimum and maximum values of x can be found by examining the endpoints of the range -π and π: x_min = cos(3*(-π)) - π = -π - 1 x_max = cos(3*(π)) + π = π + 1

Step 2: Analyze the y equation

Similarly, for the y equation, y = cos(4t) + t, we need to find the minimum and maximum values of y for the given range of t (-π ≤ t ≤ π). The cos(4t) term ranges from -1 to 1. Therefore, the minimum and maximum values of y can be found by examining the endpoints of the range -π and π: y_min = cos(4*(-π)) - π = -π - 1 y_max = cos(4*(π)) + π = π + 1

Step 3: Define the viewing window

Using the minimum and maximum values for x and y, we can define the proper viewing window for the graph of the given parametric curve. The viewing window should include the x range and y range: x: [-π-1, π+1] y: [-π-1, π+1] This window will give a complete view of the graph for the parametric curve.

Key Concepts

Graphing Parametric Curves

Parametric equations involve using one or more variables to express a set of quantities as explicit functions. In simpler terms, each parameter, usually denoted as \( t \), defines the set of both \( x \) and \( y \) in a graph.To graph a parametric curve like this one, which defines the curve through \( x = \cos(3t) + t \) and \( y = \cos(4t) + t \), you'll map the values of \( t \) within the given interval, in this case, \(-\pi \leq t \leq \pi\). The idea is to evaluate and plot the coordinates generated by these parametric equations on an \( xy \)-plane.

• Both expressions \( \cos(3t) + t \) and \( \cos(4t) + t \) modify the initial trigonometric function, adding variation by incorporating the variable \( t \) directly in the linear part.
• The periodic nature of the cosine functions creates an interesting pattern of oscillations throughout the given range. The challenge is to find the perfect window, or graphing span, to capture all of these changes.

The main task when working with parametric equations is finding a complete and clear view of how \( x \) and \( y \) interact as \( t \) varies. Understanding how these components work together helps in visualizing and solving many rich and complex problems in mathematics.

Viewing Window for Graphs

The viewing window is crucial when displaying parametric curves, as it determines what part of the graph is visible and directly affects interpretability. A poor choice might miss out on critical portions of the curve, leaving you without a complete picture.In this case, the problem was addressed by first determining the extreme points for both \( x \) and \( y \), as given by the parameter \( t \). The range \(-\pi \leq t \leq \pi\) is standard for examining the behaviors of periodic functions like cosine.

• To define the horizontal axis \( (x) \), compute the endpoints using \( x = \cos(3t) + t \). For our range, these endpoints are \( x_{min} = -\pi - 1 \) and \( x_{max} = \pi + 1 \).
• Similarly, calculate the vertical axis \( (y) \) using \( y = \cos(4t) + t \), resulting in \( y_{min} = -\pi - 1 \) and \( y_{max} = \pi + 1 \).

By choosing a viewing window that spans from \([-\pi - 1, \pi + 1]\) for both \( x \) and \( y \), the full range of the curve is visible. This ensures no part of the pattern escapes our attention, allowing us to observe all symmetrical and unique movements of the trigonometric functions combined with the parameter changes.

Analyzing Trigonometric Functions

Trigonometric functions, with their characteristic waves, play a vital role in parametric equations, adding complexity with periodic behavior. In the context of \( x = \cos(3t) + t \) and \( y = \cos(4t) + t \), understanding this periodicity is key.These cosine terms create repetitive wave patterns, which can produce beautiful and intricate graphs known as Lissajous figures when combined in parametric forms.

• For \( x = \cos(3t) + t \), a cycle of 3 periods within \( -\pi \leq t \leq \pi \) introduces an offset that translates into a unique path on the graph.
• The function \( y = \cos(4t) + t \) moves slightly differently, with 4 cycles, upping the overall complexity and interaction within the given interval.

The work with these functions requires careful handling to account for shifts and combined effects. Analyzing such behavior involves recognizing oscillation patterns and their impact on the overall design of the parametric curve. Understanding these intricacies helps students link algebraic expressions to visual representations, a valuable skill in fields like physics and engineering.