Problem 77 - Calculus Early Transcendentals

Question

How are the graphs of \(r=1+\sin (\theta-\pi / 6)\) and \(r=1+\sin (\theta-\pi / 3)\) related to the graph of \(r=1+\sin \theta\) ? In general, how is the graph of \(r=f(\theta-\alpha)\) related to the graph of \(r=f(\theta) ?\)

Step-by-Step Solution

Answer

The graphs are rotations of each other; the shift in angle results in a counterclockwise rotation by that angle.

Step 1: Understanding the Problem

We need to explore how shifting the angle in a polar function affects its graph. We are given three equations in polar coordinates and are asked to compare their graphs: 1) \(r=1+\sin \theta\), 2) \(r=1+\sin (\theta-\pi / 6)\), and 3) \(r=1+\sin (\theta-\pi / 3)\). We'll also consider the general case for any function \(r=f(\theta-\alpha)\).

Step 2: Recognizing the Effect of Angle Shifts

An angle shift of \(\alpha\) in the function \(f(\theta)\) to \(f(\theta-\alpha)\) corresponds to rotating the graph of \(f(\theta)\) by \(\alpha\) radians counterclockwise. Thus, \(r=f(\theta-\alpha)\) is the graph of \(r=f(\theta)\) rotated counterclockwise by \(\alpha\).

Step 3: Applying the Concept to the Given Functions

For the first function, \(r=1+\sin \theta\), it's a standard limaçon. For \(r=1+\sin (\theta-\pi/6)\), this graph is the original graph \(r=1+\sin \theta\) rotated by \(\pi/6\) radians counterclockwise. Similarly, \(r=1+\sin (\theta-\pi/3)\) is the original graph \(r=1+\sin \theta\) rotated by \(\pi/3\) radians counterclockwise.

Step 4: General Statement about Angle Shifts

In general, when we have a polar graph \(r=f(\theta)\) and substitute \(\theta - \alpha\) for \(\theta\), the resultant graph \(r=f(\theta-\alpha)\) is the graph of \(r=f(\theta)\), rotated by \(\alpha\) radians counterclockwise. This concept helps in understanding how the points on the graph are re-positioned on the polar plane after applying an angular shift.

Key Concepts

Angle Rotation

In polar coordinates, angle rotation plays a crucial role in how functions are graphed. Normally, a polar graph is expressed in terms of a radius, \( r \), and an angle, \( \theta \). When we alter this angle by subtracting a constant \( \alpha \), as in \( r = f(\theta - \alpha) \), the entire graph rotates counterclockwise by \( \alpha \) radians.
This is because the new angle considers the original angle shifted back by \( \alpha \) radians. As a result, each point on the graph is repositioned around the origin by this angular offset.

Key points about angle rotation in polar graphs include:

• The direction of the rotation is always counterclockwise.
• The amount of rotation is directly determined by the magnitude of \( \alpha \) in radians.
• This transformation affects the orientation but not the shape of the graph.

Understanding how shifting \( \theta \) affects a graph is essential for visualizing these transformations in polar plots.

Limaçon

A limaçon is a type of polar graph that has unique and interesting shapes. It’s usually defined by equations like \( r = 1 + \sin \theta \) or \( r = 1 + \cos \theta \). These graphs can resemble different shapes, such as cardioids or dimpled curves, which depend on the parameters used within the equation.
For the specific function \( r = 1 + \sin \theta \), the limaçon might display symmetrical, looped patterns, centered around the origin.

Various characteristics of a limaçon include:

• The presence of loops or dimples based on the coefficients in the equation.
• Symmetry about certain axes, depending on terms like \( \sin \theta \) or \( \cos \theta \).
• Shape fluctuations that occur as the angle \( \theta \) rotates.

Recognizing such properties allows for better manipulation and understanding of how these beautiful curves take form through different equations and transformations.

Graph Transformation

Graph transformation in polar coordinates encompasses the entire process of modifying a graph through changes in its underlying function. When working with polar equations, transformations can include shifts, rotations, and alterations of the graph's basic shape.
For instance, the transformation from \( r = 1 + \sin \theta \) to \( r = 1+ \sin(\theta - \pi/6) \) shifts the original graph by \( \pi/6 \) radians, creating a noticeable rotation on the polar plane.

Important aspects of graph transformations in this context are:

• Identifying the type of transformation applied—such as a rotation or shift.
• Understanding how the graph's orientation and position are affected without altering its shape.
• Using transformations to compare graphs and determine their relationships.
• Visualizing how these transformations manifest on the polar grid.

Grasping these concepts of graph transformations helps in developing a deeper intuition for analyzing and predicting changes in polar graph behaviors.