Polar Coordinates & Polar Functions (College Board AP® Precalculus): Exam Questions

What are the real part and the imaginary part of the complex number $z=-3+5i$?

The polar function $f$ is given by $f\left(\theta \right)=4\sin \theta +1$. What is the average rate of change of $r=f\left(\theta \right)$ with respect to $\theta$ on the interval $\left[0,\frac{\pi }{2}\right]$?

The table gives values for the polar function $r=f\left(\theta \right)$ at selected values of $\theta$. Assume that between any two consecutive values of $\theta$ listed in the table, $f$ is either always increasing or always decreasing.

Based on the table, at which value of $\theta$ does the graph of $r=f\left(\theta \right)$ have a point that is relatively closest to the origin?

What are the rectangular coordinates of the point with polar coordinates $\left(4,\frac{5\pi }{6}\right)$?

A complex number is given by $z=-3+4i$ and is represented by a point in the complex plane. What is the distance between this point and the origin?

The table gives values for a polar function $r=f\left(\theta \right)$ at selected values of $\theta$.

What is the average rate of change of $r$ with respect to $\theta$ on the interval $\left[0,\frac{\pi }{2}\right]$?

Which of the following polar equations represents a circle of radius $4$ centered at the origin?

The polar function $f$ has values $f\left(\frac{\pi }{6}\right)=2$ and $f\left(\frac{\pi }{2}\right)=8$. Use the average rate of change of $f$ on the interval $\left[\frac{\pi }{6},\frac{\pi }{2}\right]$ to estimate the value of $f\left(\frac{\pi }{3}\right)$.

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates $\left(-2,2\sqrt{3}\right)$. Which of the following is one way to express the complex number using its polar coordinates $\left(r,\theta \right)$ ?

The figure shows the graph of the polar function $r=f\left(\theta \right)$, where $f\left(\theta \right)=3\sin \left(2\theta \right)$, in the polar coordinate system for $0\le \theta \le 2\pi$. There are five points labeled $A$, $B$, $C$, $D$, and $E$. If the domain of $f$ is restricted to $0\le \theta \le \pi$, the portion of the given graph that remains consists of two pieces. One of those pieces is the portion of the graph in Quadrant I from $B$ through $A$ and back to $B$. Which of the following describes the other remaining piece?

Consider the graph of the polar function $r=f\left(\theta \right)$, where $f\left(\theta \right)=2-3\cos \theta$, in the polar coordinate system for $0\le \theta \le 2\pi$. Which of the following statements is true about the distance between the point with polar coordinates $\left(f\left(\theta \right),\theta \right)$ and the origin?

Which of the following could be the graph of the polar curve with equation $r=3+2\sin \theta$ in the polar coordinate plane?

A polar function is given by $r=\sin \theta -1$. As $\theta$ increases over the interval $\left(\frac{\pi }{2},\pi \right)$, which of the following describes the behavior of the distance from the pole and the position of the corresponding points relative to the polar axis?

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates $\left(-\sqrt{3},1\right)$. Which of the following is one way to express the complex number using its polar coordinates $\left(r,\theta \right)$?

A point $P$ has polar coordinates $\left(3,\frac{\pi }{4}\right)$. Which of the following is also a polar representation of $P$?

A complex number $z$ is given in polar form by $z=4\cos \left(\frac{2\pi }{3}\right)+i\left(4\sin \left(\frac{2\pi }{3}\right)\right)$. What is the rectangular form $a+bi$ of $z$?

The polar function $f$ is given by $f\left(\theta \right)=2\sin \theta +1$. Which of the following is true about the graph of $r=f\left(\theta \right)$ on the interval $0\le \theta \le 2\pi$?

The figure shows the graph of a polar function $r=f\left(\theta \right)$ in the polar coordinate system for $0\le \theta \le 2\pi$.

Which of the following could be an expression for $f\left(\theta \right)$?

The figure shows the graph of $r=f\left(\theta \right)$, where $f\left(\theta \right)=1+\cos \theta$, in the polar coordinate system for $0\le \theta \le 2\pi$.

On which of the following intervals does the graph appear in Quadrant IV?

The figure shows the graph of $r=f\left(\theta \right)$, where $f\left(\theta \right)=1+2\sin \theta$, in the polar coordinate system for $0\le \theta \le 2\pi$.

Which of the following statements is true about the distance between the point with polar coordinates $\left(f\left(\theta \right),\theta \right)$ and the origin?

The polar function $f$ is given by $f\left(\theta \right)=-2+\sin \theta$ for $0\le \theta \le 2\pi$. At which value of $\theta$ does the graph of $r=f\left(\theta \right)$ have a point that is relatively closest to the origin?

A point $P$ has polar coordinates $\left(-2,\frac{5\pi }{6}\right)$. What are the rectangular coordinates of $P$?

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates $\left(-2,-2\right)$. Which of the following is one way to express the complex number using its polar coordinates $\left(r,\theta \right)$?

Which of the following is the graph of the polar function $r=f\left(\theta \right)$, where $f\left(\theta \right)=1-2\cos \theta$, in the polar coordinate system for $0\le \theta \le 2\pi$?

Two complex numbers $z_1=\sqrt{7}+3i$ and $z_2=-2+3i$ are represented by points in the complex plane. Which of the following correctly compares the distance from $z_1$ to the origin and the distance from $z_2$ to the origin?

A polar function $f$ is given by $f\left(\theta \right)=1+2\sin \theta$ for $0\le \theta \le 2\pi$. At which value of $\theta$ is the distance between the point with polar coordinates $\left(f\left(\theta \right),\theta \right)$ and the origin the greatest?

Consider the graph of $r=f\left(\theta \right)$, where $f\left(\theta \right)=2\sin \theta -1$, in the polar coordinate system for $0\le \theta \le 2\pi$. On which of the following intervals does the value of $r$ increase while the distance between the point with polar coordinates $\left(f\left(\theta \right),\theta \right)$ and the origin decreases?

A polar function is given by $r=f\left(\theta \right)$, where $f\left(\theta \right)=2\cos \theta -1$, in the polar coordinate system for $0\le \theta \le 2\pi$. On which of the following intervals is the average rate of change of $r$ with respect to $\theta$ equal to $0$?

A polar function $r=f\left(\theta \right)$ is given by $f\left(\theta \right)=4\sin \left(3\theta \right)$, in the polar coordinate system for $0\le \theta \le 2\pi$. The graph is a rose curve. Which of the following describes the graph?

The complex number $z=-2+2\sqrt{3} i$ is written in polar form as $z=\left(r\cos \theta \right)+i\left(r\sin \theta \right)$, where $r>0$ and $0\le \theta <2\pi$. What is the value of $\theta$?

Consider the polar function $r=1-2\cos \theta$. On which of the following intervals is the distance between the graph of the function and the origin decreasing?