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

Precalculus Polar Parametric Test Answer Key: questions 1–5

Question 1: Polar to rectangular (2 points)

1. \(x=8\cos\frac{5\pi}{6}=-4\sqrt{3}\), \(y=8\sin\frac{5\pi}{6}=4\): \((-4\sqrt{3},4)\)
2. \(x=3\sqrt{2}\cdot\frac{\sqrt{2}}{2}=3\), \(y=3\sqrt{2}\cdot(-\frac{\sqrt{2}}{2})=-3\): \((3,-3)\)

Scoring: 1 point for each correct point; both coordinates must be correct.

Question 2: Rectangular to polar (2 points)

1. \(r=\sqrt{3+1}=2\); \(\tan\theta=-\frac{1}{\sqrt{3}}\) in Quadrant II, so \(\theta=\frac{5\pi}{6}\): \((2,\frac{5\pi}{6})\)
2. The point is on the negative y-axis: \((7,\frac{3\pi}{2})\)

Scoring: 1 point for each correct pair (correct r and correct angle).

Question 3: Convert equations (2 points)

1. \(x^{2}+y^{2}=4x\) gives \(r^{2}=4r\cos\theta\): \(r=4\cos\theta\)
2. \(r^{2}=-6r\cos\theta\) gives \(x^{2}+y^{2}=-6x\): \((x+3)^{2}+y^{2}=9\)

Scoring: 1 point for each equation; an equivalent form such as \(x^{2}+y^{2}=-6x\) earns full credit.

Question 4: Identify a rose (2 points)

1. 4 petals
2. The tips lie on the diagonals, not on the axes, so the curve uses sine: \(r=3\sin 2\theta\) (4 petals needs \(n=2\), length \(a=3\)).

Scoring: 1 point for the petal count; 1 point for the equation with \(a=3\) and \(n=2\) and sine.

Question 5: Analyze a limaçon (2 points)

1. \(a=1< b=2\): limaçon with an inner loop
2. \(r=3\), \(r=1\), \(r=-1\): 3, 1, -1

Scoring: 1 point for the type; 1 point for all three values (award 0 if one value is wrong).

Precalculus Polar Parametric Test Answer Key: questions 6–9

Question 6: Eliminate the parameter (2 points)

1. \(t=\frac{x-3}{2}\), so \(y=\frac{(x-3)^{2}}{4}\)
2. \(x=-1\), \(y=4\): \((-1,4)\). Check: \(\frac{(-4)^{2}}{4}=4\).

Scoring: 1 point for each part.

Question 7: Parametric circle (3 points)

1. \(\cos t=\frac{x+1}{2}\), \(\sin t=\frac{y-3}{2}\), so \((x+1)^{2}+(y-3)^{2}=4\)
2. Center \((-1,3)\), radius 2
3. A circle through \((1,3)\), \((-1,5)\), \((-3,3)\), \((-1,1)\), traced counterclockwise from \((1,3)\).

Scoring: 1 point for each part; in part (c) the circle and the direction must both be shown.

Question 8: Projectile motion (3 points)

1. \(48t-16t^{2}=16t(3-t)=0\): \(t=3\) s
2. \(x=15(3)=\) 45 ft
3. The vertex is at \(t=1.5\): \(y=72-36=\) 36 ft

Scoring: 1 point for each part.

Question 9: Two hikers (2 points)

1. The rays differ by \(75^{\circ}-15^{\circ}=60^{\circ}\). \(d^{2}=36+64-2(6)(8)\cos 60^{\circ}=100-48=52\): \(d=2\sqrt{13}\) mi
2. \(\approx 7.21\) mi

Scoring: 1 point for each part.

Scoring summary

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