Problem 4 - Precalculus

Question

Graph the polar equations. $$r=\theta / \pi, \text { for }-4 \pi \leq \theta \leq 0$$

Step-by-Step Solution

Answer

The graph is a spiral that moves inward from radius \(-4\) to \(0\) as \(\theta\) increases from \(-4\pi\) to \(0\).

Step 1: Identify the Equation

The given equation is a polar equation: \( r = \frac{\theta}{\pi} \). Polar equations relate the radius \( r \) and angle \( \theta \) in polar coordinates.

Step 2: Determine the Range for θ

The problem specifies the range of the angle \( \theta \) as \(-4\pi \leq \theta \leq 0\). This range indicates that the graph should cover angles from \(-4\pi\) to \(0\).

Step 3: Substitute Values into Equation

Substitute various values within the specified range into the equation \( r = \frac{\theta}{\pi} \). For instance, if \( \theta = -4\pi \), then \( r = -4 \), and if \( \theta = 0 \), then \( r = 0 \).

Step 4: Plotting Points in Polar Coordinates

Plot each pair \((\theta, r)\) from the substitutions onto the polar coordinate system. Remember that each \( \theta \) is an angle, and \( r \) is the distance from the origin.

Step 5: Connect the Dots and Analyze

Connect the plotted points to reveal the graph's shape. For our equation, the graph will spiral inward from a radius of \(-4\) at \(\theta = -4\pi\) to a radius of \(0\) at \(\theta = 0\), forming a counterclockwise arch.

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative way to describe points in a plane, often used for graphing equations that relate a radius and angle. In the polar coordinate system, a point is defined by two values: the angle, typically denoted as \( \theta \), and the radius, denoted as \( r \).

Unlike Cartesian coordinates, where points are determined by distances along perpendicular axes, polar coordinates rely on rotational distances from the origin (also known as the pole).

• \( \theta \): This is the angle measured counterclockwise from the positive x-axis. It is typically given in radians.
• \( r \): This is the radial distance from the origin to the point. It can be positive or negative, affecting how the point plots with respect to \( \theta \).

Polar coordinates facilitate the expression and graphing of certain curves and shapes that are more cumbersome in Cartesian coordinates.

Graphing in Polar Coordinates

Graphing in polar coordinates involves plotting points based on their radii and angles. This method aids in visualizing relationships in polar equations. To graph an equation such as \( r = \frac{\theta}{\pi} \), follow these steps:

1. **Identify the Range**: Establish your range for \( \theta \) as given in the problem, for instance, from \(-4\pi\) to \(0\). This range will dictate the portion of the graph you will analyze.
2. **Compute Points**: Calculate \( r \) for various \( \theta \) within the range. Substitute these angles into the equation. For example, if \( \theta = -2\pi \), then \( r = -2 \).

3. **Plot Points**: On polar graph paper, locate each point using its \( \theta \) to find the angle and \( r \) for the distance from the origin.
4. **Connect the Points**: Line segments can join successive points to illustrate the curve, revealing the shape dictated by the polar equation.

Graphing in polar coordinates requires interpreting both \( \theta \) and \( r \) to accurately plot and connect all the points.

Radius and Angle Relationship

The radius and angle relationship in polar equations plays a crucial role in defining the graph's shape. The equation \( r = \frac{\theta}{\pi} \) ties together the radius, \( r \), with the angle, \( \theta \), indicating a direct proportional relationship. As \( \theta \) increases or decreases, \( r \) does too, reflecting a dynamic, spiral-like graph.

Understanding this relationship helps with predicting how changes in angle impact the graph:

• **Direct Proportion**: As \( \theta \) changes within its range, \( r \) changes proportionally. If \( \theta = -4\pi \), the equation results in \( r = -4 \), explaining a wide spread on the graph.
• **Graph Direction**: Negative values of \( r \) signify plotting in the opposite direction from the angle's axis, describing shapes like spirals or loops in some cases.
• **Parameters**: Since \( \theta \) varies from \(-4\pi\) to \(0\), \( r \) follows suit, transitioning from negative to zero as the graph progresses counterclockwise.

Recognizing how \( \theta \) and \( r \) integrate supports graph interpretation and understanding their contribution to the curve's structure.