Polar Coordinates and Parametric Equations: Precalculus Math Test – PDF Download

Question 1: Polar to rectangular (2 points)

Convert each polar point to rectangular coordinates.

1. \((8,\frac{5\pi}{6})\) (1 point)
2. \((3\sqrt{2},\frac{7\pi}{4})\) (1 point)

Question 2: Rectangular to polar (2 points)

Convert each point to polar coordinates with \(r> 0\) and \(0\le\theta< 2\pi\).

1. \((-\sqrt{3},1)\) (1 point)
2. \((0,-7)\) (1 point)

Question 3: Convert equations (2 points)

Convert each equation to the other coordinate system.

1. Write \(x^{2}+y^{2}-4x=0\) in polar form. (1 point)
2. Write \(r=-6\cos\theta\) in rectangular form. (1 point)

Question 4: Identify a rose (2 points)

Use the graph of a rose curve \(r=a\sin n\theta\) or \(r=a\cos n\theta\). Each petal tip is 3 units from the pole.

1. How many petals does the graph have? (1 point)
2. Write an equation for the graph. (1 point)

Question 5: Analyze a limaçon (2 points)

Answer each part for \(r=1+2\cos\theta\).

1. Name the type of graph. (1 point)
2. Find \(r\) when \(\theta=0\), \(\theta=\frac{\pi}{2}\) and \(\theta=\pi\). (1 point)

Question 6: Eliminate the parameter (2 points)

Answer each part for a point that moves according to \(x=2t+3\), \(y=t^{2}\), where \(t\) is any real number.

1. Eliminate the parameter to write \(y\) in terms of \(x\). (1 point)
2. Find the point when \(t=-2\). (1 point)

Question 7: Parametric circle (3 points)

Answer each part for a point that moves according to \(x=-1+2\cos t\), \(y=3+2\sin t\), \(0\le t\le 2\pi\).

1. Eliminate the parameter. (1 point)
2. State the center and radius. (1 point)
3. Sketch the path on the grid and show its direction. (1 point)

Question 8: Projectile motion (3 points)

Answer each part for a ball thrown by Noah. Its position after \(t\) seconds is \(x=15t\), \(y=48t-16t^{2}\), in feet.

1. Find the time the ball lands. (1 point)
2. Find how far it travels horizontally. (1 point)
3. Find its maximum height. (1 point)

Question 9: Two hikers (2 points)

Answer each part for a ranger station at the pole. Hiker 1 is at \((6\text{ mi},15^{\circ})\) and hiker 2 is at \((8\text{ mi},75^{\circ})\).

1. Find the exact distance between the hikers. (1 point)
2. Find that distance to the nearest hundredth of a mile. (1 point)

See every chapter on the Precalculus math page.