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Graphing Other Common Polar Equations quiz

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  • What are the four common shapes of polar equations discussed in this lesson?
    Cardioids, limaçons, roses, and lemniscates.
  • How can you distinguish a cardioid from a limaçon based on their equations?
    Both have equations of the form r = a ± b cos(θ) or r = a ± b sin(θ), but for a cardioid, a = b; for a limaçon, a ≠ b.
  • What determines whether a limaçon has a dimple or an inner loop?
    If a > b, the limaçon has a dimple; if a < b, it has an inner loop.
  • What is the general equation for a rose curve in polar coordinates?
    r = a cos(nθ) or r = a sin(nθ), where a ≠ 0 and n is an integer ≥ 2.
  • How do you determine the number of petals in a rose curve?
    If n is even, the rose has 2n petals; if n is odd, it has n petals.
  • What is the unique feature of a lemniscate equation compared to other polar equations?
    Lemniscate equations contain r², such as r² = ±a² cos(2θ) or r² = ±a² sin(2θ).
  • How do you classify the equation r = 1 + cos(θ)?
    It is a cardioid because a and b are both 1 and the equation uses addition.
  • What symmetry does a cardioid with a cosine function have?
    It is symmetric about the polar axis.
  • When graphing a cardioid, which angles should you plot points at?
    Plot points at the quadrantal angles: 0, π/2, π, and 3π/2.
  • How do you determine the symmetry of a limaçon with a sine function?
    It is symmetric about the line θ = π/2.
  • What does the equation r = 3 - 2 sin(θ) represent and why?
    It represents a limaçon with a dimple because a = 3 > b = 2 and uses subtraction.
  • How do you find the spacing between petals in a rose curve?
    Divide 2π by the number of petals to get the angular spacing.
  • For the equation r = 4 cos(2θ), how many petals does the rose have and where is the first petal?
    It has 4 petals (since n = 2, 2n = 4), and the first petal is at θ = 0.
  • How do you find the value of 'a' in a lemniscate equation like r² = 4 sin(2θ)?
    Take the square root of 4 to get a = 2.
  • Where is the first petal of the lemniscate r² = 4 sin(2θ) located, and how do you find the second?
    The first petal is at θ = π/4; the second is found by reflecting over the pole.