Precalcus written test.

MATH 2412 ~ PreCalculus Online

Chapter 11 Test (includes 2.2, Parametric Equations, Conics in Polar Form) ~ Form S

Answer each question carefully and completely showing as much work as possible. For multi-stepped problems you must show your work in order to receive full credit. Using only a calculator to find the answer is not sufficient. In this case, you must use algebra to prove your answer is correct. Unless stated otherwise, the calculator should only be used to check your answer. Make sure that all answers are in simplest form and that you use the correct symbols. If indicated, you must use the requested method in order to receive full credit.

SHOW ALL WORK ON SEPARATE PAPER.

IT IS A GOOD IDEA TO USE GRAPH PAPER FOR ALL GRAPHS.

1. Find the center and radius of the circle with the given equation. Then sketch the graph.

Include and clearly label the center and at least four points on the graph.

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2. Find the a) vertex, b) focus, and c) directrix of the parabola below. Then sketch the graph. Include and clearly label all three on the graph.

�� � �� � �� �� 3. A driving light has a parabolic cross section with a depth of 2 inches at its center and 4 inches

in diameter at the top. Where should the light source be placed to maximize the output of illumination?

4. Find the a) center, b) vertices and c) foci of the following ellipse then sketch the graph. Include and clearly label all on the graph.

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5. An elliptical arch railroad tunnel 12 feet high at the center and 28 feet wide is cut through the

side of a mountain. Find the equation of the ellipse if the endpoint of the minor axis is represented by the point (0, 12).

6. Find the a) center, b) vertices, c) foci, d) endpoints of the conjugate axis, and e) the

equations of the asymptotes of the hyperbola below. Then sketch the graph. Include and clearly label all on the graph.

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7. A nuclear power plant has a large cooling tower with sides curved in the shape of a hyperbola.

The radius of the base of the tower is 80 meters. The radius of the top of the tower is 70 meters. The sides of the tower are 40 meters apart at the closest point located 150 meters above the ground.

a. Find the equation of the hyperbola that describes the sides of the cooling tower.

Assume that the center is at the origin. b. Determine the height of the tower.

8. Find the eccentricity, equation of the directrix, and any foci of the following conic section. Then

sketch the graph. Be sure to explain your answer completely and carefully. Include and label at least 4 points on the graph.

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9. Let � � � � and � ����. a. Sketch the graph of the curve described by the parametric equations, showing the

direction (with arrows) of increasing values of t. Show your work.

b. Find a corresponding rectangular equation. 10. Find parametric equations for the curve satisfying the given conditions. Give your answers

such that x and y are functions of t and neither is x = t or y = t.

A parabola with vertex at (1, -9) and passing through (0, 0) and (2, 0).