Types of Graphs (Polar Graph) Flashcards

Questions and Answers

What is the definition of a Line in polar coordinates?

θ = Angle

r = secθ

r = cscθ

All of the above (correct)

What is the polar equation of a Circle?

r = a (radius a, centered at origin/pole)

What is the polar equation of a Lemniscate?

r² = a²sin(2θ) or r² = a²cos(2θ)

What is the equation of the Spiral of Archimedes?

r = θ Signup and view all the answers

What condition defines an Inner Loop Limaçon?

|a| < |b| Signup and view all the answers

How is a Cardioid Limaçon defined?

|a| = |b| Signup and view all the answers

What defines a Dimpled Limaçon?

|a| > |b| Signup and view all the answers

What defines a Convex Limaçon?

|a| > |2b| Signup and view all the answers

What is the polar equation for a Rose curve with n odd?

r = acos(nθ) or r = asin(nθ) Signup and view all the answers

What does the polar coordinate 'x' represent?

rcosθ Signup and view all the answers

What does the polar coordinate 'y' represent?

rsinθ Signup and view all the answers

What is the equation for r² in polar coordinates?

x² + y² Signup and view all the answers

What is the relationship between y and x in polar coordinates?

tanθ = y/x Signup and view all the answers

Study Notes

Polar Graph Basics

Polar coordinates use the angle (θ) and the radial distance (r) to define points in a plane.
Key relationships include r = secθ and r = cscθ associated with lines.

Circle Characteristics

Circle defined by r = a, where 'a' is the radius centered at the origin.
Symmetry principles:
- r = a cosθ shows symmetry about the x-axis (θ = 0°).
- r = a sinθ shows symmetry about the y-axis (θ = 90°).

Lemniscate Properties

Defined by equations r² = a²sin(2θ) and r² = a²cos(2θ).
Exhibits symmetry with respect to the origin, with petal length equal to 'a'.

Spiral of Archimedes

Represented by the equation r = θ.
This graph operates in radian mode and exhibits a continuous outward spiral.

Limaçon Variants

Inner Loop Limaçon: occurs when |a| < |b|, leading to a shape with an inner loop.
Cardioid Limaçon: characterized by |a| = |b|, forming a heart shape.
Dimpled Limaçon: presents when |a| > |b|, with softer curves resembling dimples.
Convex Limaçon: exists when |a| > |2b|, providing a convex outline without loops.

Rose Curves

Defined by r = a cos(nθ) and r = a sin(nθ).
Symmetry types depend on 'n':
- Even n results in 2n petals; odd n gives n petals.
- Symmetrical about the x-axis, y-axis, and the origin.

Converting Polar to Cartesian Coordinates

The conversion to Cartesian coordinates includes:
- x = r cos(θ)
- y = r sin(θ)
The relation between polar radius and Cartesian coordinates: r² = x² + y².
The angle tangent relationship in polar: tan(θ) = y/x.

Description

Explore the key concepts and definitions related to polar graphs through this interactive flashcard quiz. Learn about different types of graphs such as lines, circles, lemniscates, and spirals, including their equations and properties. Perfect for reinforcing your understanding of polar coordinates.