High school calculus quiz: Graphing common polar equations

📚 Quick Study Guide

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• Polar Coordinates: A point in the polar coordinate system is represented by $(r, \theta)$, where $r$ is the distance from the origin (pole) and $\theta$ is the angle from the positive x-axis (polar axis).
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• Basic Polar Equations: Common polar equations include lines ($\theta = a$), circles ($r = a$), and more complex curves like cardioids and roses.
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• Converting to Cartesian: You can convert from polar to Cartesian coordinates using the relationships: $x = r \cos(\theta)$ and $y = r \sin(\theta)$.
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• Symmetry Tests: Tests for symmetry can simplify graphing. For example, if replacing $\theta$ with $-\theta$ doesn't change the equation, the graph is symmetric about the x-axis.
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• Roses: Equations of the form $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$ create rose curves. If $n$ is even, the rose has $2n$ petals; if $n$ is odd, it has $n$ petals.
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• Cardioids: Equations of the form $r = a(1 \pm \cos(\theta))$ or $r = a(1 \pm \sin(\theta))$ create cardioids (heart-shaped curves).

Practice Quiz

1. What type of curve is represented by the polar equation $r = 5$?
1. A) A line
2. B) A circle
3. C) A spiral
4. D) A cardioid
2. Which of the following polar equations represents a vertical line?
1. A) $r = 5 \cos(\theta)$
2. B) $\theta = \frac{\pi}{2}$
3. C) $r = 5 \sin(\theta)$
4. D) $r = 5$
3. How many petals does the rose curve $r = 3 \cos(2\theta)$ have?
1. A) 2
2. B) 3
3. C) 4
4. D) 6
4. What is the shape of the polar equation $r = 2(1 + \cos(\theta))$?
1. A) A lemniscate
2. B) A rose
3. C) A circle
4. D) A cardioid
5. Which test can be used to check for symmetry about the y-axis in a polar equation?
1. A) Replacing $r$ with $-r$
2. B) Replacing $\theta$ with $-\theta$
3. C) Replacing $(r, \theta)$ with $(-r, -\theta)$
4. D) Replacing $\theta$ with $\pi - \theta$
6. What is the maximum value of $r$ for the polar equation $r = 4 \sin(3\theta)$?
1. A) 1
2. B) 3
3. C) 4
4. D) 12
7. Which polar equation represents a circle centered at the origin with radius 7?
1. A) $\theta = 7$
2. B) $r = 7 \cos(\theta)$
3. C) $r = 7 \sin(\theta)$
4. D) $r = 7$