Polar Function Graphs - AP Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

9. What is the polar axis in the polar coordinate system?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

37. What is the pole in the polar coordinate system?

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

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

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

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

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

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

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

Study these flashcards