9.2.1: Resources and Key Concepts
- Page ID
- 196994
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Prerequisite Topics
This page lists the prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.
- Systems
- Solving Systems of Nonlinear Equations: Finding the intersection points of two polar curves is equivalent to solving a system of nonlinear equations.
Videos
- Polar Graphs
- Graphs of Polar Equations - The General Idea Behind Graphing Polar Curves
- Graphs of Polar Equations - Graphing Polar Functions of the Form \( r(\theta)=A \sin(\theta) \) or \( r(\theta)=A \cos(\theta) \)
- Graphs of Polar Equations - Graphing Polar Functions of the Form \( r(\theta)=A \pm B \sin(\theta) \) or \( r(\theta)=A \pm B \cos(\theta) \)
- Graphs of Polar Equations - Graphing Polar Functions of the Form \( r(\theta)=A \sin(B \theta) \) or \( r(\theta)=A \cos(B \theta) \)
- Graphs of Polar Equations - Graphing Polar Functions of the Form \( r(\theta) = c \) or \( \theta = c \)
- Graphs of Polar Equations - Naming the Polar Curves
Key Concepts
Definitions
- This section focuses on graphing polar equations and does not introduce any new vocabulary terms. The names of various polar curves (circles, roses, cardioids, etc.) are introduced as categories of graphs.
Common Mistakes
- Incorrectly Tracing the Graph: When plotting points for a polar graph, it is essential to connect them in order of increasing \(\theta\). Failing to do so can result in an incorrect shape, especially for curves like limaçons and roses that may trace over themselves or have loops.
- Missing Intersection Points at the Pole: When finding the intersection of two polar graphs \(r = f(\theta)\) and \(r = g(\theta)\), solving \(f(\theta) = g(\theta)\) may not reveal an intersection at the pole. This is because the pole can be represented by \(r=0\) for many different values of \(\theta\). You must always check separately if the pole (\(r=0\)) is a point on both graphs.
- Assuming a Single Representation: A single polar graph can often be represented by multiple different equations. For example, a circle centered on the y-axis is typically written with sine, but a phase-shifted cosine function could produce the same graph.