Problem 71 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=-6(1+\cos \theta)$$

Step-by-Step Solution

Answer

The graph is a limacon with an inner loop, symmetric about the polar axis.

Step 1: Identify the Type of Polar Equation

Recognize that the given equation \( r = -6(1 + \cos \theta) \) is of the form \( r = a(1 + \cos \theta) \). This type of equation represents a conic section in polar coordinates, specifically a limacon.

Step 2: Analyze the Equation Components

The equation has the form \( r = -6(1 + \cos \theta) \). Here, \(a = -6\). Since \(a\) is negative, this indicates that the limacon will have an inner loop. Also, since the equation includes \( \cos \theta\), it is symmetric with respect to the polar axis (horizontal axis).

Step 3: Determine the Inner and Outer Loop

The inner loop occurs when \( r \) changes sign. With \( a = -6 \), the maximum radius \( r \) is \( -6 imes 2 = -12 \) at \( \theta = 0 \) (which is a reflection across the origin in actual graphing). The smallest magnitude should be \( r = 0 \), which occurs at \( \theta = \pi \). This indicates the presence of an inner loop.

Step 4: Identify Key Points for Graphing

Calculate key points for different angles to sketch the graph. For example:- When \( \theta = 0 \), \( r = -6(1 + 1) = -12 \).- When \( \theta = \pi \), \( r = -6(1 - 1) = 0 \).- When \( \theta = \frac{\pi}{2} \) or \( \theta = \frac{3\pi}{2} \), \( r = -6 \).

Step 5: Sketch the Graph

Begin by plotting the key points in polar coordinates. At \( \theta = 0 \) and \( \theta = 2\pi \), the point is at \( -12 \) (through origin, off-page to opposite side). At \( \theta = \pi \), \( r = 0 \), so mark the origin. Between these key angles, calculate points for smooth curve transition.

Key Concepts

Limacon

A limacon (pronounced "lee-mah-son") is a unique type of polar curve derived from the equation \( r = a(1 + heta) \) or \( r = a(1 - heta) \), where \( a \) is a constant. These curves are particularly notable for their versatile shapes, ranging from heart-like figures to loops. The nature of the curve changes dramatically based on certain attributes of the equation.

• When \( a \) is positive: If \( a \) is positive, the curves can depict a dimpled shape or cusp."
• Negative \( a \): A curve where \( a \) is negative, as in our exercise, reflects through the origin. If the absolute value of \( a \) is greater than 1, the curve typically forms a loop, called an inner loop.

In context, for our equation \( r = -6(1 + \ \cos \ \theta) \), the negative value of \( a \) indicates the shape will be flipped through the origin, forming an inner loop as depicted by one of these configurations. In sketching limacons, recognizing the role of \( a \) can greatly simplify predicting the curve's characteristic behavior.

Conic Section

Conic sections are the curves obtained by slicing a cone at various angles. This concept traditionally produces circular, elliptical, parabolic, and hyperbolic shapes. However, in polar coordinates, limacons can be considered an offshoot or variation of these classic conic figures.

• Connection to Limacons: In polar coordinates, limacons are uniquely identified, belonging to a special category that includes a blend of some characteristics from circles or even parts of ellipses (yet appear distinct due to their characteristic loops or dimples).
• Symmetry: They are symmetrical respective to the horizontal polar axis due to the presence of terms like \(\cos \ \theta\).
• Transformation: Recognizing these transforms in polar equations allows one to recognize varieties of conic sections even amidst the complex translations that define polar curves.

This background enriches understanding of how such polar-based forms relate to their conic section origins, enabling deeper appreciation when studying their graphing.

Graphing Polar Coordinates

Graphing in polar coordinates can initially appear daunting, but thrillingly, it offers a nuanced perspective on geometry. In polar coordinates, curves are plotted based on angle \(\theta\) and radius \(r\), differing significantly from the conventional Cartesian approach.

• Origin of Graphing: The origin acts differently here. In our given exercise for \( r = -6(1 + \cos \ \theta) \), the negative sign indicates portions of the graph will extend across the origin.
• Key Points: Identifying essential points is crucial. For example, when \(\theta = 0\), the equation produces \( r = -12 \). At \(\theta = \pi\), \(r = 0\) gives direct evidence of points marked at the origin.
• Smooth Curves: Between these noticeable points, it’s essential to determine additional values of \(\theta\) which help create a smooth transition of curves for accuracy.

To adeptly sketch polar graphs, understanding step-by-step transitions between each angle ensures not only a clearer graph but insights into the interplay of algebraic symmetry involved.