Polar Function Graphs - AP Precalculus
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What symmetry test indicates a polar graph is symmetric about the line $\theta=\frac{\pi}{2}$?
What symmetry test indicates a polar graph is symmetric about the line $\theta=\frac{\pi}{2}$?
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Replace $\theta$ with $\pi-\theta$; equation unchanged. Reflection across $y$-axis supplements angle to $\pi$.
Replace $\theta$ with $\pi-\theta$; equation unchanged. Reflection across $y$-axis supplements angle to $\pi$.
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What is the polar equation of the $x$-axis?
What is the polar equation of the $x$-axis?
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$\theta=0$. Horizontal line through origin has angle $0°$ from positive $x$-axis.
$\theta=0$. Horizontal line through origin has angle $0°$ from positive $x$-axis.
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What symmetry test indicates a polar graph is symmetric about the polar axis?
What symmetry test indicates a polar graph is symmetric about the polar axis?
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Replace $\theta$ with $-\theta$; equation unchanged. Reflection across $x$-axis negates angle.
Replace $\theta$ with $-\theta$; equation unchanged. Reflection across $x$-axis negates angle.
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What is the polar equation of the $y$-axis?
What is the polar equation of the $y$-axis?
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$\theta=\frac{\pi}{2}$. Vertical line through origin has angle $90°$ from positive $x$-axis.
$\theta=\frac{\pi}{2}$. Vertical line through origin has angle $90°$ from positive $x$-axis.
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What is the conversion from polar to Cartesian coordinates for a point $(r,\theta)$?
What is the conversion from polar to Cartesian coordinates for a point $(r,\theta)$?
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$x=r\cos\theta,\ y=r\sin\theta$. Uses trigonometric definitions where $x$ is horizontal and $y$ is vertical projection.
$x=r\cos\theta,\ y=r\sin\theta$. Uses trigonometric definitions where $x$ is horizontal and $y$ is vertical projection.
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What is the conversion from Cartesian to polar coordinates for a point $(x,y)$?
What is the conversion from Cartesian to polar coordinates for a point $(x,y)$?
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$r=\sqrt{x^2+y^2},\ \tan\theta=\frac{y}{x}$. Distance formula gives $r$; angle found from slope ratio.
$r=\sqrt{x^2+y^2},\ \tan\theta=\frac{y}{x}$. Distance formula gives $r$; angle found from slope ratio.
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What is the equation relating $r$, $x$, and $y$ for polar coordinates?
What is the equation relating $r$, $x$, and $y$ for polar coordinates?
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$r^2=x^2+y^2$. Pythagorean theorem relates radius to Cartesian coordinates.
$r^2=x^2+y^2$. Pythagorean theorem relates radius to Cartesian coordinates.
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Identify the polar equation of a circle centered at the origin with radius $a>0$.
Identify the polar equation of a circle centered at the origin with radius $a>0$.
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$r=a$. Constant radius from origin defines a circle.
$r=a$. Constant radius from origin defines a circle.
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What is the polar equation of the line through the origin making angle $\alpha$ with the positive $x$-axis?
What is the polar equation of the line through the origin making angle $\alpha$ with the positive $x$-axis?
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$\theta=\alpha$. Constant angle creates a ray from the origin.
$\theta=\alpha$. Constant angle creates a ray from the origin.
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What symmetry test indicates a polar graph is symmetric about the pole (origin)?
What symmetry test indicates a polar graph is symmetric about the pole (origin)?
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Replace $r$ with $-r$ (or $\theta$ with $\theta+\pi$); unchanged. Point and its opposite have same location.
Replace $r$ with $-r$ (or $\theta$ with $\theta+\pi$); unchanged. Point and its opposite have same location.
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Identify the period of $r=\cos\theta$ and $r=\sin\theta$ as polar functions.
Identify the period of $r=\cos\theta$ and $r=\sin\theta$ as polar functions.
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Period $=2\pi$. Basic trig functions complete one cycle in $2\pi$.
Period $=2\pi$. Basic trig functions complete one cycle in $2\pi$.
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Identify the period of $r=\cos(n\theta)$ and $r=\sin(n\theta)$ for integer $n\neq 0$.
Identify the period of $r=\cos(n\theta)$ and $r=\sin(n\theta)$ for integer $n\neq 0$.
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Period $=\frac{2\pi}{|n|}$. Coefficient $n$ compresses period by factor of $|n|$.
Period $=\frac{2\pi}{|n|}$. Coefficient $n$ compresses period by factor of $|n|$.
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How many petals does the rose curve $r=a\cos(n\theta)$ have when $n$ is odd?
How many petals does the rose curve $r=a\cos(n\theta)$ have when $n$ is odd?
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$n$ petals. Odd $n$ traces all petals in one period.
$n$ petals. Odd $n$ traces all petals in one period.
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How many petals does the rose curve $r=a\cos(n\theta)$ have when $n$ is even?
How many petals does the rose curve $r=a\cos(n\theta)$ have when $n$ is even?
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$2n$ petals. Even $n$ requires two periods to trace all petals.
$2n$ petals. Even $n$ requires two periods to trace all petals.
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What is the maximum value of $r$ for $r=a\cos\theta$ (or $r=a\sin\theta$) with $a>0$?
What is the maximum value of $r$ for $r=a\cos\theta$ (or $r=a\sin\theta$) with $a>0$?
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$r_{\max}=a$. Cosine/sine maximum is 1, scaled by coefficient $a$.
$r_{\max}=a$. Cosine/sine maximum is 1, scaled by coefficient $a$.
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Find the polar intercept angles where $r=a\cos\theta$ crosses the pole (origin).
Find the polar intercept angles where $r=a\cos\theta$ crosses the pole (origin).
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$\theta=\frac{\pi}{2},\ \frac{3\pi}{2}$. $r=0$ when $\cos\theta=0$, at odd multiples of $\frac{\pi}{2}$.
$\theta=\frac{\pi}{2},\ \frac{3\pi}{2}$. $r=0$ when $\cos\theta=0$, at odd multiples of $\frac{\pi}{2}$.
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Find the polar intercept angles where $r=a\sin\theta$ crosses the pole (origin).
Find the polar intercept angles where $r=a\sin\theta$ crosses the pole (origin).
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$\theta=0,\ \pi$. $r=0$ when $\sin\theta=0$, at multiples of $\pi$.
$\theta=0,\ \pi$. $r=0$ when $\sin\theta=0$, at multiples of $\pi$.
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Convert the polar point $(r,\theta)=(2,\frac{\pi}{3})$ to Cartesian coordinates.
Convert the polar point $(r,\theta)=(2,\frac{\pi}{3})$ to Cartesian coordinates.
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$(x,y)=(1,\sqrt{3})$. $x=2\cos(\frac{\pi}{3})=1$, $y=2\sin(\frac{\pi}{3})=\sqrt{3}$.
$(x,y)=(1,\sqrt{3})$. $x=2\cos(\frac{\pi}{3})=1$, $y=2\sin(\frac{\pi}{3})=\sqrt{3}$.
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Convert the Cartesian point $(x,y)=(-\sqrt{3},1)$ to polar with $r>0$ and $0\le\theta<2\pi$.
Convert the Cartesian point $(x,y)=(-\sqrt{3},1)$ to polar with $r>0$ and $0\le\theta<2\pi$.
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$(r,\theta)=(2,\frac{5\pi}{6})$. $r=\sqrt{3+1}=2$; $\theta$ in Q2 where $\tan\theta=-\frac{1}{\sqrt{3}}$.
$(r,\theta)=(2,\frac{5\pi}{6})$. $r=\sqrt{3+1}=2$; $\theta$ in Q2 where $\tan\theta=-\frac{1}{\sqrt{3}}$.
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Identify the equivalent polar coordinate of $(r,\theta)=(3,\frac{\pi}{6})$ using a negative radius.
Identify the equivalent polar coordinate of $(r,\theta)=(3,\frac{\pi}{6})$ using a negative radius.
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$(-3,\frac{7\pi}{6})$. Add $\pi$ to angle and negate radius for equivalent point.
$(-3,\frac{7\pi}{6})$. Add $\pi$ to angle and negate radius for equivalent point.
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What is the symmetry test for a polar equation using the pole (origin)?
What is the symmetry test for a polar equation using the pole (origin)?
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Replace $r$ with $-r$ or $\theta$ with $\theta+\pi$; same implies symmetry. Points symmetric about pole are $\pi$ radians apart.
Replace $r$ with $-r$ or $\theta$ with $\theta+\pi$; same implies symmetry. Points symmetric about pole are $\pi$ radians apart.
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Which condition determines a limacon $r=a+b\cos(\theta)$ has an inner loop?
Which condition determines a limacon $r=a+b\cos(\theta)$ has an inner loop?
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Inner loop occurs when $|a|<|b|$. Inner loop forms when $r$ becomes negative for some $\theta$.
Inner loop occurs when $|a|<|b|$. Inner loop forms when $r$ becomes negative for some $\theta$.
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What is the graph type of $r=a(1+\cos(\theta))$ for $a>0$?
What is the graph type of $r=a(1+\cos(\theta))$ for $a>0$?
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Cardioid (a limacon with a cusp). Heart-shaped curve with cusp at pole when $\theta=\pi$.
Cardioid (a limacon with a cusp). Heart-shaped curve with cusp at pole when $\theta=\pi$.
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For a rose $r=a\cos(n\theta)$, what is the maximum radius (petal length)?
For a rose $r=a\cos(n\theta)$, what is the maximum radius (petal length)?
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Maximum $r=|a|$. Petals reach farthest when $\cos(n\theta)=\pm 1$.
Maximum $r=|a|$. Petals reach farthest when $\cos(n\theta)=\pm 1$.
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For a rose $r=a\sin(n\theta)$ with even $n$, how many petals are graphed?
For a rose $r=a\sin(n\theta)$ with even $n$, how many petals are graphed?
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$2n$ petals. Even $n$ requires full period $[0,2\pi]$ to trace all petals.
$2n$ petals. Even $n$ requires full period $[0,2\pi]$ to trace all petals.
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For a rose $r=a\cos(n\theta)$ with odd $n$, how many petals are graphed?
For a rose $r=a\cos(n\theta)$ with odd $n$, how many petals are graphed?
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$n$ petals. Odd $n$ traces the complete rose in one period $[0,\pi]$.
$n$ petals. Odd $n$ traces the complete rose in one period $[0,\pi]$.
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What circle does $r=4\sin(\theta)$ represent (center and radius)?
What circle does $r=4\sin(\theta)$ represent (center and radius)?
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Center $(0,2)$, radius $2$. Complete the square: $x^2+(y-2)^2=4$ from $x^2+y^2=4y$.
Center $(0,2)$, radius $2$. Complete the square: $x^2+(y-2)^2=4$ from $x^2+y^2=4y$.
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What circle does $r=2\cos(\theta)$ represent (center and radius)?
What circle does $r=2\cos(\theta)$ represent (center and radius)?
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Center $(1,0)$, radius $1$. Complete the square: $(x-1)^2+y^2=1$ from $x^2+y^2=2x$.
Center $(1,0)$, radius $1$. Complete the square: $(x-1)^2+y^2=1$ from $x^2+y^2=2x$.
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What is the Cartesian form of the polar equation $r=2\sin(\theta)$?
What is the Cartesian form of the polar equation $r=2\sin(\theta)$?
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$x^2+y^2=2y$. Substitute $r\sin(\theta)=y$ and $r^2=x^2+y^2$ to convert.
$x^2+y^2=2y$. Substitute $r\sin(\theta)=y$ and $r^2=x^2+y^2$ to convert.
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What is the Cartesian form of the polar equation $r=2\cos(\theta)$?
What is the Cartesian form of the polar equation $r=2\cos(\theta)$?
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$x^2+y^2=2x$. Substitute $r\cos(\theta)=x$ and $r^2=x^2+y^2$ to convert.
$x^2+y^2=2x$. Substitute $r\cos(\theta)=x$ and $r^2=x^2+y^2$ to convert.
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