Unit 7 test: Polar and Parametric Equations

Convert the polar point $(6, \tfrac{7\pi}{4})$ to rectangular coordinates.

Write the rectangular point $(-\sqrt{3}, 1)$ in polar form $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$.

Which polar point names the same point as $(-5, \tfrac{\pi}{3})$?

Find the exact distance between the polar points $(4, \tfrac{\pi}{3})$ and $(6, \pi)$.

How many petals does the rose $r = 5\cos(5\theta)$ have?

The polar equation $r = -8\cos\theta$ is a circle. Find its center in rectangular coordinates.

Which best describes the graph of $r = 2 + 4\cos\theta$?

Convert the polar equation $r = \dfrac{5}{\sin\theta - 2\cos\theta}$ to a rectangular equation. Solve for $y$.

Find the argument $\theta$ of $z = -2 - 2\sqrt{3}\,i$, with $0 \le \theta < 2\pi$.

Let $z_1 = 4\left(\cos\tfrac{2\pi}{3} + i\sin\tfrac{2\pi}{3}\right)$ and $z_2 = 2\left(\cos\tfrac{5\pi}{6} + i\sin\tfrac{5\pi}{6}\right)$. Write $z_1 z_2$ in the form $a + bi$ and enter $(a, b)$.

Use De Moivre's Theorem to evaluate $\left(\sqrt{3} + i\right)^6$.

Which number is a fourth root of $-16$?

Eliminate the parameter from $x = t - 1$, $y = t^2 + 3$. Write $y$ as a function of $x$.

The segment from $(-3, 4)$ to $(5, 0)$ is parametrized by $x = -3 + 8t$, $y = 4 - 4t$, $0 \le t \le 1$. Find the point where $t = \tfrac{3}{4}$.

A ball is thrown so that $x = 30t$ and $y = 6 + 40t - 16t^2$, with distances in feet and $t$ in seconds. What is the ball's maximum height, in feet?