Problem 74 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=1+2 \cos \theta$$

Step-by-Step Solution

Answer

Graph the limaçon symmetric about the polar axis with a loop.

Step 1: Understand the Polar Equation

The given polar equation is \( r = 1 + 2 \cos \theta \). This type of equation typically describes a limaçon, a family of heart-shaped curves on the polar coordinate plane.

Step 2: Identify Symmetry

The equation includes a \( \cos \theta \) term, indicating that the graph will be symmetric with respect to the polar axis (the horizontal axis). This symmetry will be evident in how the shape develops as \( \theta \) varies.

Step 3: Find Key Points

Calculate \( r \) for important values of \( \theta \) to understand the shape. For example:- At \( \theta = 0\), \( r = 1 + 2 \cdot 1 = 3 \).- At \( \theta = \frac{\pi}{2}\), \( r = 1 + 0 = 1 \).- At \( \theta = \pi\), \( r = 1 + 2(-1) = -1 \), which reflects through the origin.- At \( \theta = \frac{3\pi}{2}\), \( r = 1 \).

Step 4: Shape Analysis and Range of \(r\)

The value of \( r \) ranges from 1 to 3 when \( \theta \) varies from 0 to \( \pi \), and it transforms back from 1 to -1 from \( \pi \) to \( 2\pi \), demonstrating the contour of a limaçon with a loop.

Step 5: Sketch the Graph

Draw the polar coordinate system. Start by plotting key points for \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) and gradually plot intermediate points to capture the curve smoothly. Reflect around the origin when \( r \) becomes negative to form the inner loop.

Key Concepts

Limaçon

In polar graphs, curves resembling a heart shape are often called limaçons. The name comes from the French word for "snail." The polar equation \( r = 1 + 2 \cos \theta \) is an example of a limaçon with an inner loop. This type of limaçon occurs when the coefficient of \( \cos \theta \) is greater than the constant term. Here, 2 is greater than 1, which leads to the formation of a loop.

A limaçon's shape can vary depending on the ratio between the constant and the coefficient of \( \cos \theta \) or \( \sin \theta \).

• An inner loop forms when the coefficient is larger.
• A dimpled (without a loop) limaçon happens when the constant is larger.
• A cardioid forms when the coefficients are equal.

In understanding a limaçon, recognizing these variations helps predict the graph's structure based on the given polar equation.

Polar Coordinates

Polar coordinates are essential for depicting curves like limaçons. Unlike the Cartesian coordinate system which uses \(x\) and \(y\) positions on a grid, polar coordinates use \(r\) (radius) and \(\theta\) (angle).

• The radius \(r\) measures the distance from the origin.
• The angle \(\theta\) indicates direction, measured in radians or degrees from a reference direction.

With the given equation \( r = 1 + 2 \cos \theta \), various \(\theta\) values will generate elliptic paths as \( r \) adjusts based on the cosine function.
This system naturally describes circles, spirals, and other curves by showing how far and in what direction you move from a central point.
In practice, the value of \( \theta \) is adjusted through its typical range of \( 0 \) to \( 2\pi \) (or 0 to 360 degrees), to map out the full path of the graph.

Symmetry in Polar Graphs

Understanding symmetry in polar graphs assists in predicting and sketching complex shapes like limaçons. The equation \( r = 1 + 2 \cos \theta \) reveals that the graph will be symmetric along the polar axis.

This symmetry allows us to simplify sketching:

• If you know one side of the axis, you can reflect it to form the other side.
• For \( \cos \) terms, symmetry typically lies on the horizontal or polar axis.
• For \( \sin \) terms, symmetry would align with the vertical axis.

Leveraging symmetry reduces the calculation needed to understand the overall shape, providing a clearer and quicker insight into the graph's structure.
This particular equation's polar axis symmetry ensures predictability; visually appealing yet mathematically consistent.