# M52 Practice Final. = 4y focus: 0, 1 ˆ Ë. y focus: 0, Name: Class: Date:

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1 Class: Date: M52 Practice Final Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the standard form of the equation of the parabola and determine the coordinates of the focus. = y focus: 0, 1 ˆ Á = 1 1 ˆ y focus: 0, 16 Á 16 = y focus: ( 0, ) = 1 y focus: 0, 1 ˆ Á = 1 1 ˆ y focus: 0, Á Find the standard form of the equation of the parabola with the given characteristic and vertex at the origin. focus: (0, 7) y 2 = 28x = 28y = 7y y 2 = 7x = 7y 1

2 3. Find the standard form of the equation of the parabola with the given characteristic and vertex at the origin. directrix: x = y y 2 = x = y = y y 2 = x. Find the vertex and focus of the parabol y 2 = 8 x vertex: (0, 0) focus: Á 32,0 ˆ vertex: (0, 0) focus: 0, ˆ Á 8 vertex: 0, 5 ˆ Á focus: 0, ˆ Á 32 vertex: (0, 0) focus: Á 8,0 ˆ vertex: 0, 5 ˆ Á focus: 8, ˆ Á 8 5. Find the vertex and focus of the parabol + 8y = 0 vertex: ( 2, 0) focus: (0, 0) vertex: (0, 0) focus: ( 2, 0) vertex: (0, 0) focus: ( 0, 2) vertex: (0, 0) focus: ( 0, 2) vertex: ( 2, 0) focus: (0, 0) 2

3 6. Match the equation with its graph. + y 2 = 3

4 7. Find the vertex and focus of the parabol Á y 2 ˆ ( x 3) = 0 vertex: ( 3, 2) focus: ( 3, 1) vertex: ( 3, 2) focus: ( 7, 2) vertex: ( 3, 2) focus: ( 3, 18) vertex: ( 3,2) focus: ( 3, 2) vertex: ( 3,2) focus: ( 1, 2) 8. Find the vertex and directrix of the parabol + 2x y + 17 = 0 vertex: ( 1, ) directrix: y = 5 vertex: ( 1, ) directrix: y = 5 vertex: ( 1, ) directrix: y = 3 vertex: ( 1, ) directrix: y = 2 vertex: ( 1, ) directrix: y = 0. Give the standard form of the equation of the parabola with the given characteristics. vertex: ( 3, 1) focus: ( 1, 1) Á y + 1 ˆ 2 = 8( x 3) Á y 1 ˆ 2 = 8( x + 3) ( x 3) = 8 Á y + 1 ˆ Á y 1 ˆ 2 = 8( x + 3) ( x + 3) = 8 Á y 1 ˆ 10. Give the standard form of the equation of the parabola with the given characteristics. vertex: ( 1, 3) directrix: x = 6 ( x 1) = 20 Á y 3ˆ Á y 3 ˆ 2 = 20( x 1) Á y + 3 ˆ 2 = 20( x + 1) Á y 3 ˆ 2 = 20( x 1) ( x + 1) = 20 Á y + 3ˆ

5 11. Give the standard form of the equation of the parabola with the given characteristics. focus: ( 1, 0) directrix: y = Á y + 2 ˆ 2 = 8( x + 1) ( x + 2) = 8 Á y + 1 ˆ Á y 2 ˆ 2 = 8( x 1) ( x + 1) 2 = 8 Á y + 2ˆ ( x 1) 2 = 8 Á y 2ˆ 12. A solar oven uses a parabolic reflector to focus the sun's rays at a point 6 inches from the vertex of the reflector (see figure). Write an equation for a cross section of the oven's reflector with its focus on the positive y axis and its vertex at the origin. y = 2 y 6 = 2y y = 6 = 6y L = 6 inches 5

6 13. An elliptical stained-glass insert is to be fitted in a ftby6ft rectangular opening (see figure). Using the coordinate system shown, find an equation for the ellips + y2 y2 2 y2 3 y2 + y2 1. Find the standard form of the equation of the ellipse with the following characteristics. foci: ( ú, 0) major axis of length: y y y y y

7 15. Find the standard form of the equation of the ellipse with the given characteristics. vertices: (, 0), (, 16) minor axis of length x y 82 x y + 2 x y 82 6 x y 2 x 2 + y Find the standard form of the equation of the ellipse with the following characteristics. foci: ( ±, 0) major axis of length: y y y y y2 36 7

8 17. Find the standard form of the equation of the ellipse with the given characteristics. center: ( 7,6) a = 5c foci: ( 3,6), ( 11, 6) x y x y x y x y x y Find the standard form of the equation of the ellipse with the given characteristics. foci: (, 3),,13 ( ) endpoints of the major axis: (, 0),,16 ( ) x y x y 82 3 x y x y 82 6 x y 2 3 8

9 1. Find the standard form of the equation of the ellipse with the given characteristics. foci: (, ), (, 3) endpoints of the major axis: (, 1), (, 2) ( x + ) Á y + 6 ˆ ( x ) Á y 6 ˆ ( x 6) Á y ˆ ( x + 6) Á y + ˆ ( x + ) Á y + 6 ˆ Find the center and vertices of the ellips + y2 center: (7, 2) vertices: ( 7, 2), (7, 2) center: (0, 0) vertices: (0, 7), (0, 7) center: (0, 0) vertices: ( 7, 0), (7, 0) center: (7, 0) vertices: (0, 2), (0, 2) center: (0, 0) vertices: ( 2, 0), (2, 0) 21. Find the center and foci of the ellips ( x + 5) (y + )2 + 5 center: ( 5,) foci: ( 5, 7), ( 5,11) center: ( 5, ) foci: ( 7, ), ( 3, ) center: ( 5, ) foci: ( 5, 11), ( 5, 7) center: ( 5, ) foci: ( 3, ), ( 7,) center: ( 5,) foci: ( 3, ), ( 7, )

10 22. Identify the conic by writing the equation in standard form. 10y y + 160x 5 = 0 Á y 3ˆ Á y + 3ˆ Á y 3ˆ Á y + 3ˆ Á y + 3ˆ ( x ) ( x )2 7 ( x )2 5 ( x )2 5 + ( x ) ; ellipse ; hyperbola ; hyperbola ; hyperbola ; ellipse 23. Identify the conic by writing the equation in standard form. + y 2 + 0x + 16y + 0 = 0 ( x + 5) 2 Á y + 2 ˆ 2 + ; ellipse 6 6 ( x + 5) 2 + Á y + 2ˆ 2 = 3; circle ( x + 5) 2 + Á y + 2ˆ 2 ; circle ( x + 5) 2 + Áy + 2ˆ 2 = 2; circle ( x + 5) Á y + 2 ˆ 2 ; ellipse 11 10

11 2. Find the center and vertices of the ellips + y 2 2x + 72y + 1 = 0 center: (, 3) vertices: ( 7, 3), ( 1,3) center: ( 3, ) vertices: ( 0, ), ( 6, ) center: ( 3, ) vertices: ( 5, ), ( 1,) center: ( 3, ) vertices: ( 6,), ( 0,) center: ( 3, ) vertices: ( 1, ), ( 5, ). Identify the conic by writing the equation in standard form. + y 2 + 0x + 8y + 13 = 0 ( x + 5) 2 Á y + 1 ˆ 2 + ; ellipse ( 3x + 5) 2 + Á2y + 1 ˆ 2 = 167; circle ( x + 5) 2 Á y + 1 ˆ 2 + ; ellipse ( x + 5) 2 Á y + 1 ˆ 2 + ; ellipse Á y + 1 ˆ 2 ( x + 5) ; ellipse 26. Find the center and the vertices of the ellips 16 + y 2 = 6 center: ( 8, 8) vertices: ( 2, ), (2, ) center: (0, 0) vertices: (0, ), (0, ) center: (0, 0) vertices: (, 2), (, 2) center: (0, 0) vertices: (, 0), (, 0) center: ( 8, 8) vertices: ( 2, 0), (2, 0) 27. Find the center and foci of the ellips ( x 8) (y 2) center: ( 8, 2) foci: ( 8, 6), ( 8, 2) center: (8, 2) foci: ( 12, 2), (, 2) center: ( 8, 2) foci: ( 12, 2), (, 2) center: (8, 2) foci: (8, 2), (8, 6) center: (8, 2) foci: (, 2), (12, 2) 11

12 28. Find the center and vertices of the ellips + y x 5y = 0 center: (3, 8) vertices: (0, 8), (6, 8) center: ( 8, 3) vertices: ( 11, 3), ( 5, 3) center: (8, 3) vertices: (7, 3), (, 3) center: (8, 3) vertices: (5, 3), (11, 3) center: ( 8, 3) vertices: (, 3), ( 7, 3) 2. Find the standard form of the equation of the ellipse with vertices ( ±7, 0) and eccentricity e = 7. + y y2 + y y2 16 y2 30. Find the standard form of the equation of the ellipse with vertices ( 0, ±5) and eccentricity e = y2 16 y2 + y2 + y y2 12

13 31. Find the vertices and asymptotes of the hyperbol y 2 16 vertices: ( 0, ±) asymptote: y = ± 3 x vertices: ( ±, 3) asymptote: y = ± 3 x vertices: ( ±, 0) asymptote: y = ± 3 x vertices: ( ±, 0) asymptote: y = ± 3 x vertices: ( 0, ±) asymptote: y = ± 3 x 32. Find the vertices and asymptotes of the hyperbol 36 + y2 vertices: ( ±6, 0) asymptote: y = ± 6 7 x vertices: ( 0, ±6) asymptote: y = ± 7 6 x vertices: ( ±6, 0) asymptote: y = ± 7 6 x vertices: ( ±6, 7) asymptote: y = ± 6 7 x vertices: ( 0, ±6) asymptote: y = ± 6 7 x 33. Find the center and foci of the hyperbol Á y + ˆ x + 1) center: ( 1, ), foci: (, ), (7, ) center: ( 1, ), foci: ( 1, 12), ( 1, ) center: (, 1), foci: (, 7), (, ) center: (, 1), foci: ( 12, 1), (, 1) center: (1, ), foci: (1, ), (1, 12) 3. Find the center and vertices of the hyperbol 11 y x + 0y 88 = 0 center: ( 1, 5), vertices: ( 1, 0), ( 1, 10) center: (1, 5), vertices: (1, 10), (1, 0) center: ( 1, 5), vertices: ( 6, 5), (,5) center: ( 5, 1), vertices: ( 10, 1), (0, 1) center: (1, 5), vertices: (, 5), (6, 5) 13

14 35. Find the standard form of the equation of the hyperbola with the given characteristics. vertices: ( 0, ±6) foci: ( 0, ±7) y y y2 13 y = 36 y2 13 = 36. Find the standard form of the equation of the hyperbola with the given characteristics. vertices: ( 2, ), ( 2, 6) foci: ( 2, 5), ( 2, 7) Á y 1ˆ 2 Á y 1ˆ 2 Á y 2ˆ 2 11 Á y + 1ˆ 2 Á y + 1ˆ 2 ( x + 2)2 11 ( x + 2)2 36 ( x + 1)2 ( x 2)2 11 ( x 2)2 36 1

15 37. Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (0, 1), (10, 1) asymptotes: y = 3 5 x, y = 3 5 x Á y + 5ˆ 2 ( x + 5) 2 Á y 1ˆ 2 Á y + 1ˆ 2 ( x 5) 2 ( x 1)2 Á y 1 ˆ 2 ( x + 5)2 ( x 5)2 Á y + 1 ˆ Find the standard form of the equation of the hyperbola with the given characteristics. foci: ( ±, 0) asymptotes: y = ±5x 16 y2 y 2 16 y y y

16 M52 Practice Final Answer Section MULTIPLE CHOICE 1. ANS: E PTS: 1 REF: 213 OBJ: Find the standard form of a parabola with a vertex at the origin 2. ANS: B PTS: 1 REF: 211 OBJ: Find the standard form of a parabola with a vertex at the origin 3. ANS: B PTS: 1 REF: 212 OBJ: Find the standard form of a parabola with a vertex at the origin. ANS: A PTS: 1 REF: 20 OBJ: Find the vertex and focus of a parabola 5. ANS: D PTS: 1 REF: 210 OBJ: Find the vertex and focus of a parabola 6. ANS: B PTS: 1 REF: 208 OBJ: Identify graphs of conic sections 7. ANS: E PTS: 1 REF: 23 OBJ: Find the vertex and focus of a parabola 8. ANS: C PTS: 1 REF: 235 OBJ: Find the vertex and focus of a parabola. ANS: B PTS: 1 REF: 238 OBJ: Find the standard form of a parabola 10. ANS: C PTS: 1 REF: 23 OBJ: Find the standard form of a parabola 11. ANS: D PTS: 1 REF: 20 OBJ: Find the standard form of a parabola 12. ANS: C PTS: 1 REF: 21 OBJ: Find equations of parabolas in everyday situations 13. ANS: A PTS: 1 REF: 220 OBJ: Find equations of ellipses in everyday situations 1. ANS: A PTS: ANS: C PTS: ANS: A PTS: 1 REF: 21 OBJ: Find the standard form of a ellipse with a vertex at the origin 17. ANS: C PTS: ANS: D PTS: 1 1. ANS: E PTS: 1 REF: 28 OBJ: Find equation of ellipse using center and foci 20. ANS: C PTS: 1 REF: 216 OBJ: Find center and vertices of an ellipse centered at the origin 21. ANS: C PTS: 1 REF: 2 OBJ: Find center and foci of an ellipse 22. ANS: D PTS: ANS: C PTS: 1 2. ANS: B PTS: 1 REF: OBJ: Find center and vertices of an ellipse. ANS: A PTS: 1 1

17 26. ANS: B PTS: 1 REF: 217 OBJ: Find center and vertices of an ellipse centered at the origin 27. ANS: D PTS: ANS: B PTS: 1 2. ANS: C PTS: 1 REF: 2 OBJ: Find the standard form of an ellipse using eccentricity 30. ANS: C PTS: 1 REF: 0 OBJ: Find the standard form of an ellipse using eccentricity 31. ANS: E PTS: 1 REF: 2 OBJ: Find vertices and asymptotes of a hyperbola centered at the origin 32. ANS: C PTS: 1 REF: 22 OBJ: Find vertices and asymptotes of a hyperbola centered at the origin 33. ANS: B PTS: 1 3. ANS: C PTS: ANS: A PTS: 1 REF: 226 OBJ: Find the standard form of the equation of a hyperbola centered at the origin 36. ANS: A PTS: ANS: E PTS: ANS: D PTS: 1 REF: 227 OBJ: Find the standard form of the equation of a hyperbola centered at the origin 2

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