Types of Graphs (Polar Graph) Flashcards

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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?

<p>r = θ</p> Signup and view all the answers

What condition defines an Inner Loop Limaçon?

<p>|a| &lt; |b| (B)</p> Signup and view all the answers

How is a Cardioid Limaçon defined?

<p>|a| = |b| (A)</p> Signup and view all the answers

What defines a Dimpled Limaçon?

<p>|a| &gt; |b| (A)</p> Signup and view all the answers

What defines a Convex Limaçon?

<p>|a| &gt; |2b| (A)</p> Signup and view all the answers

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

<p>r = acos(nθ) or r = asin(nθ)</p> Signup and view all the answers

What does the polar coordinate 'x' represent?

<p>rcosθ</p> Signup and view all the answers

What does the polar coordinate 'y' represent?

<p>rsinθ</p> Signup and view all the answers

What is the equation for r² in polar coordinates?

<p>x² + y²</p> Signup and view all the answers

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

<p>tanθ = y/x</p> Signup and view all the answers

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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.

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