AP Precalculus Flashcards: Polar Function Graphs

What this deck covers

This deck focuses on Polar Function Graphs, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What symmetry test indicates a polar graph is symmetric about the line $\theta=\frac{\pi}{2}$?

Answer: Replace $\theta$ with $\pi-\theta$; equation unchanged. Reflection across $y$-axis supplements angle to $\pi$.

Flashcard 2: What is the polar equation of the line through the origin making angle $\alpha$ with the positive $x$-axis?

Answer: $\theta=\alpha$. Constant angle creates a ray from the origin.

Flashcard 3: What equation relates $r$, $x$, and $y$ for polar graphs in the plane?

Answer: $r^2=x^2+y^2$. Squaring both sides of $r=\sqrt{x^2+y^2}$ gives this identity.

Flashcard 4: Convert the polar point $(r,\theta)=(2,\frac{\pi}{3})$ to Cartesian coordinates.

Answer: $(x,y)=(1,\sqrt{3})$. $x=2\cos(\frac{\pi}{3})=1$, $y=2\sin(\frac{\pi}{3})=\sqrt{3}$.

Flashcard 5: Convert the Cartesian point $(x,y)=(-\sqrt{3},1)$ to polar with $r>0$ and $0\le\theta<2\pi$.

Answer: $(r,\theta)=(2,\frac{5\pi}{6})$. $r=\sqrt{3+1}=2$; $\theta$ in Q2 where $\tan\theta=-\frac{1}{\sqrt{3}}$.

Flashcard 6: What is the equation relating $r$, $x$, and $y$ for polar coordinates?

Answer: $r^2=x^2+y^2$. Pythagorean theorem relates radius to Cartesian coordinates.

Flashcard 7: For a rose $r=a\sin(n\theta)$ with even $n$, how many petals are graphed?

Answer: $2n$ petals. Even $n$ requires full period $[0,2\pi]$ to trace all petals.

Flashcard 8: What symmetry test indicates a polar graph is symmetric about the polar axis?

Answer: Replace $\theta$ with $-\theta$; equation unchanged. Reflection across $x$-axis negates angle.

Flashcard 9: What is the polar axis in the polar coordinate system?

Answer: The positive $x$-axis. The reference line from which angles $\theta$ are measured.

Flashcard 10: How many petals does the rose curve $r=a\cos(n\theta)$ have when $n$ is even?

Answer: $2n$ petals. Even $n$ requires two periods to trace all petals.

Flashcard 11: What is the polar-to-Cartesian conversion formula for $x$ and $y$?

Answer: $x=r\cos(\theta),\ y=r\sin(\theta)$. Uses trigonometric definitions on the unit circle extended to radius $r$.

Flashcard 12: What circle does $r=2\cos(\theta)$ represent (center and radius)?

Answer: Center $(1,0)$, radius $1$. Complete the square: $(x-1)^2+y^2=1$ from $x^2+y^2=2x$.

Flashcard 13: What is the symmetry test for a polar equation using the pole (origin)?

Answer: Replace $r$ with $-r$ or $\theta$ with $\theta+\pi$; same implies symmetry. Points symmetric about pole are $\pi$ radians apart.

Flashcard 14: Identify an equivalent coordinate to $(r,\theta)$ using a $2\pi$ angle change.

Answer: $(r,\theta+2\pi k)$ for any integer $k$. Adding multiples of $2\pi$ to $\theta$ gives the same point.

Flashcard 15: Find the polar intercept angles where $r=a\cos\theta$ crosses the pole (origin).

Answer: $\theta=\frac{\pi}{2},\ \frac{3\pi}{2}$. $r=0$ when $\cos\theta=0$, at odd multiples of $\frac{\pi}{2}$.

Flashcard 16: Find the polar intercept angles where $r=a\sin\theta$ crosses the pole (origin).

Answer: $\theta=0,\ \pi$. $r=0$ when $\sin\theta=0$, at multiples of $\pi$.

Flashcard 17: Identify an equivalent coordinate to $(r,\theta)$ using a negative radius.

Answer: $(-r,\theta+\pi)$. Negative $r$ reflects the point through the pole.

Flashcard 18: Identify the equivalent polar coordinate of $(r,\theta)=(3,\frac{\pi}{6})$ using a negative radius.

Answer: $(-3,\frac{7\pi}{6})$. Add $\pi$ to angle and negate radius for equivalent point.

Flashcard 19: What symmetry test indicates a polar graph is symmetric about the pole (origin)?

Answer: Replace $r$ with $-r$ (or $\theta$ with $\theta+\pi$); unchanged. Point and its opposite have same location.

Flashcard 20: Identify the period of $r=\cos\theta$ and $r=\sin\theta$ as polar functions.

Answer: Period $=2\pi$. Basic trig functions complete one cycle in $2\pi$.

Flashcard 21: What is the conversion from polar to Cartesian coordinates for a point $(r,\theta)$?

Answer: $x=r\cos\theta,\ y=r\sin\theta$. Uses trigonometric definitions where $x$ is horizontal and $y$ is vertical projection.

Flashcard 22: Which condition determines a limacon $r=a+b\cos(\theta)$ has an inner loop?

Answer: Inner loop occurs when $|a|<|b|$. Inner loop forms when $r$ becomes negative for some $\theta$.

Flashcard 23: What circle does $r=4\sin(\theta)$ represent (center and radius)?

Answer: Center $(0,2)$, radius $2$. Complete the square: $x^2+(y-2)^2=4$ from $x^2+y^2=4y$.

Flashcard 24: What is the symmetry test for a polar equation using the $x$-axis (polar axis)?

Answer: Replace $\theta$ with $-\theta$; same equation implies symmetry. Points symmetric about polar axis have angles $\theta$ and $-\theta$.

Flashcard 25: Identify the period of $r=\cos(n\theta)$ and $r=\sin(n\theta)$ for integer $n\neq 0$.

Answer: Period $=\frac{2\pi}{|n|}$. Coefficient $n$ compresses period by factor of $|n|$.

Flashcard 26: What is the polar equation of the $y$-axis?

Answer: $\theta=\frac{\pi}{2}$. Vertical line through origin has angle $90°$ from positive $x$-axis.

Flashcard 27: What is the polar equation of the $x$-axis?

Answer: $\theta=0$. Horizontal line through origin has angle $0°$ from positive $x$-axis.

Flashcard 28: What is the graph type of $r=a$ for $a>0$ in polar coordinates?

Answer: Circle centered at the pole with radius $a$. Constant $r$ means all points are equidistant from pole.

Flashcard 29: What is the maximum value of $r$ for $r=a\cos\theta$ (or $r=a\sin\theta$) with $a>0$?

Answer: $r_{\max}=a$. Cosine/sine maximum is 1, scaled by coefficient $a$.

Flashcard 30: Identify the polar equation of a circle centered at the origin with radius $a>0$.

Answer: $r=a$. Constant radius from origin defines a circle.

Flashcard 31: What is the graph type of $r=a(1+\cos(\theta))$ for $a>0$?

Answer: Cardioid (a limacon with a cusp). Heart-shaped curve with cusp at pole when $\theta=\pi$.

Flashcard 32: What is the symmetry test for a polar equation using the $y$-axis ($\theta=\frac{\pi}{2}$ line)?

Answer: Replace $\theta$ with $\pi-\theta$; same equation implies symmetry. Points symmetric about $y$-axis have supplementary angles from polar axis.

Flashcard 33: What is the Cartesian form of the polar equation $r=2\sin(\theta)$?

Answer: $x^2+y^2=2y$. Substitute $r\sin(\theta)=y$ and $r^2=x^2+y^2$ to convert.

Flashcard 34: What is the Cartesian form of the polar equation $r=2\cos(\theta)$?

Answer: $x^2+y^2=2x$. Substitute $r\cos(\theta)=x$ and $r^2=x^2+y^2$ to convert.

Flashcard 35: For a rose $r=a\cos(n\theta)$ with odd $n$, how many petals are graphed?

Answer: $n$ petals. Odd $n$ traces the complete rose in one period $[0,\pi]$.

Flashcard 36: What is the conversion from Cartesian to polar coordinates for a point $(x,y)$?

Answer: $r=\sqrt{x^2+y^2},\ \tan\theta=\frac{y}{x}$. Distance formula gives $r$; angle found from slope ratio.

Flashcard 37: What is the pole in the polar coordinate system?

Answer: The origin $(0,0)$. The fixed point from which all distances $r$ are measured.

Flashcard 38: How many petals does the rose curve $r=a\cos(n\theta)$ have when $n$ is odd?

Answer: $n$ petals. Odd $n$ traces all petals in one period.

Flashcard 39: What is the Cartesian-to-polar conversion formula for $r$ in terms of $x$ and $y$?

Answer: $r=\sqrt{x^2+y^2}$. Apply the Pythagorean theorem to find distance from origin.

Flashcard 40: For a rose $r=a\cos(n\theta)$, what is the maximum radius (petal length)?

Answer: Maximum $r=|a|$. Petals reach farthest when $\cos(n\theta)=\pm 1$.