Skip to ContentGo to accessibility pageKeyboard shortcuts menu
Precalculus

# Practice Test

PrecalculusPractice Test

Menu

### Practice Test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

1.

$x 2 9 + y 2 4 =1 x 2 9 + y 2 4 =1$

2.

$9 y 2 +16 x 2 −36y+32x−92=0 9 y 2 +16 x 2 −36y+32x−92=0$

For the following exercises, sketch the graph, identifying the center, vertices, and foci.

3.

$( x−3 ) 2 64 + ( y−2 ) 2 36 =1 ( x−3 ) 2 64 + ( y−2 ) 2 36 =1$

4.

$2 x 2 + y 2 +8x−6y−7=0 2 x 2 + y 2 +8x−6y−7=0$

5.

Write the standard form equation of an ellipse with a center at $( 1,2 ), ( 1,2 ),$ vertex at $( 7,2 ), ( 7,2 ),$ and focus at $( 4,2 ). ( 4,2 ).$

6.

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

7.

$x 2 49 − y 2 81 =1 x 2 49 − y 2 81 =1$

8.

$16 y 2 −9 x 2 +128y+112=0 16 y 2 −9 x 2 +128y+112=0$

For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

9.

$( x−3 ) 2 25 − ( y+3 ) 2 1 =1 ( x−3 ) 2 25 − ( y+3 ) 2 1 =1$

10.

$y 2 − x 2 +4y−4x−18=0 y 2 − x 2 +4y−4x−18=0$

11.

Write the standard form equation of a hyperbola with foci at $( 1,0 ) ( 1,0 )$ and $( 1,6 ), ( 1,6 ),$ and a vertex at $( 1,2 ). ( 1,2 ).$

For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

12.

$y 2 +10x=0 y 2 +10x=0$

13.

$3 x 2 −12x−y+11=0 3 x 2 −12x−y+11=0$

For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

14.

$( x−1 ) 2 =−4( y+3 ) ( x−1 ) 2 =−4( y+3 )$

15.

$y 2 +8x−8y+40=0 y 2 +8x−8y+40=0$

16.

Write the equation of a parabola with a focus at $( 2,3 ) ( 2,3 )$ and directrix $y=−1. y=−1.$

17.

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

For the following exercises, determine which conic section is represented by the given equation, and then determine the angle $θ θ$ that will eliminate the $xy xy$ term.

18.

$3 x 2 −2xy+3 y 2 =4 3 x 2 −2xy+3 y 2 =4$

19.

$x 2 +4xy+4 y 2 +6x−8y=0 x 2 +4xy+4 y 2 +6x−8y=0$

For the following exercises, rewrite in the $x ′ y ′ x ′ y ′$ system without the $x ′ y ′ x ′ y ′$ term, and graph the rotated graph.

20.

$11 x 2 +10 3 xy+ y 2 =4 11 x 2 +10 3 xy+ y 2 =4$

21.

$16 x 2 +24xy+9 y 2 −125x=0 16 x 2 +24xy+9 y 2 −125x=0$

For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

22.

$r= 3 2−sinθ r= 3 2−sinθ$

23.

$r= 5 4+6cosθ r= 5 4+6cosθ$

For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.

24.

$r= 12 4−8sinθ r= 12 4−8sinθ$

25.

$r= 2 4+4sinθ r= 2 4+4sinθ$

26.

Find a polar equation of the conic with focus at the origin, eccentricity of $e=2, e=2,$ and directrix: $x=3. x=3.$

Do you know how you learn best?
Kinetic by OpenStax offers access to innovative study tools designed to help you maximize your learning potential.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.