1. Find the equation of the parabola with the given information. Graph youranswer.

a. Focus at (0, 0); directrix x = 4

Ans: x− 2 = −18y2

b. Vertex at (0, 0); focus at (3, 0)

Ans: x =1

12y2

c. Focus at (3, 4); vertex at (3, 2)

Ans: y − 2 =1

8(x− 3)2

d. Vertex at (0, 0); directrix y = −4

Ans: y =1

16x2

2. Find the vertex, focus, directrix, and axis of symmetry of each parabola, andgraph your answer.

a. 4x = y2 − 4y

Ans: Vertex: (−1, 2); Focus: (0, 2);directrix: x = −2; axis of symmetry: y = 2

b. x2 = y + 2x

Ans: Vertex: (1,−1); Focus:(1,−3

4

);

directrix: y = −54; axis of symmetry: x = 1

c. y2 + 6y + 8x− 7 = 0

Ans: Vertex: (2,−3); Focus: (0,−3);directrix: x = 4; axis of symmetry: y = −3

d. x2 − 6x+ 10y − 1 = 0

Ans: Vetex: (3, 1); Focus:

(3,−3

2

);

directrix: y =7

2; axis of symmetry: x = 3

e. x2 + 10x− 2y + 21 = 0

Ans: Vertex: (−5,−2); Focus:(−5,−3

2

);

directrix: y = −52; axis of symmetry: x = −5

f. x2 + 8y + 4x− 4 = 0

Ans: Vertex: (−2, 1); Focus: (−2,−1);directrix: y = 3; axis of symmetry: x = −2

3. Find the equation of the circle with the given information, and graph youranswer:

a. Center (0, 0), radius 2.

Ans: x2 + y2 = 4

b. Center (0, 0), contains the point (−3,−4).Ans: x2 + y2 = 25

c. Center (−2,−4), contains the point (1,−1).

Ans: (x+ 2)2 + (y + 4)2 = 18

d. Diameter has endpoints (3, 4) and (−1, 2).Ans: (x− 1)2 + (y − 3)2 = 5

4. Find the center and radius, and graph the given circle:

a. x2 + y2 = 25

Ans: Center = (0, 0), r = 5

b. (x+ 2)2 + (y − 3)2 = 9

Ans: Center = (−2, 3), r = 3

c. x2 − 6x+ y2 = 1

Ans: Center = (3, 0), r =√10

d. x2 − x+ y2 + y =1

2

Ans: Center =

(1

2,−1

2

), r = 1

e. x2 +1

2x+ y2 +

1

2y =

1

8

Ans: Center = −(1

4,−1

4

), r =

1

2

5. Find the center, foci, and graph the given ellipse

a. 5x2 + y2 = 25

Ans: center: (0, 0); Foci: (0,−2√5), (0, 2

√5)

b. 4x2 + 3y2 = 48

Ans: center: (0, 0); Foci: (0,−2), (0, 2)

c. 9x2 + 4y2 = 9

Ans: Center: (0, 0); Foci:

(0,−√5

2

),

(0,

√5

2

)

d.(x+ 4)2

9+

(y + 2)2

4= 1

Ans: Center: (−4,−2); Foci: (−4−√5,−2), (−4 +

√5,−2)

e. 9(x− 3)2 + (y + 2)2 = 18

Ans: Center: (3,−2); Foci: (3,−6), (3, 2)

f. x2 + 3y2 − 12y + 9 = 0

Ans: Center: (0, 2); Foci: (−√2, 2), (

√2, 2)

g. 9x2 + 4y2 − 18x+ 16y − 11 = 0

Ans: Center: (1,−2); Foci: (1,−2−√5), (1,−2 +

√5)

h. 4x2 + y2 + 4y = 0

Ans: Center: (0,−2); Foci: (0,−2−√3), (0,−2 +

√3)

6. Find the equation of the ellipse with the given information. Graph youranswer.

a. Foci at (0,±2); length of major axis is 8

Ans:x2

12+

y2

16= 1

b. Focus at (−4, 0); vertices at (±5, 0)

Ans:x2

25+

y2

9= 1

c. Focus at (0,−4); vetrices at (0,±8)

Ans:x2

48+

y2

64= 1

d. Foci at (0,±3); x−intercepts are ±2

Ans:x2

4+

y2

13= 1

e. Vertices at (±4, 0); y−intercepts are ±1

Ans:x2

16+ y2 = 1

f. Center (−3, 1); vertex (−3, 3); focus (−3, 0)

Ans:(x+ 3)2

3+

(y − 1)2

4= 1

g. Foci at (1, 2) and (−3, 2); vertex at (−4, 2)

Ans:(x+ 1)2

9+

(y − 2)2

5= 1

h. Foci at (5, 1) and (−1, 1); length of the major axis is 8

Ans:(x− 2)2

16+

(y − 1)2

7= 1

i. Center at (1, 2); focus at (1, 4); contains the point (2, 2)

Ans: (x− 1)2 +(y − 2)2

5= 1

7. Find the equation of the hyperbola with the given information. Graph youranswer.

a. Center at (0, 0); focus at (3, 0); vertex at (1, 0)

Ans: x2 − y2

8= 1

b. Focus at (0, 6); vertices at (0,−2) and (0, 2)

Ans: −x2

32+

y2

4= 1

c. Vetices at (−4, 0) and (4, 0); asymptote y = 2x

Ans:x2

16− y2

64= 1

d. Foci at (−4, 0) and (4, 0); asymptote y = −x

Ans:x2

8− y2

8= 1

e. Center at (4,−1); focus at (7,−1); vertex at (6,−1)

Ans:(x− 4)2

4− (y + 1)2

5= 1

f. Center at (−3, 1); focus at (−3, 6); vetex at (−3, 4)

Ans: −(x+ 3)2

16+

(y − 1)2

9= 1

g. Focus at (−4, 0); vertices at (−4, 4) and (−4, 2)

Ans: −(x+ 4)2

8+ (y − 3)2 = 1

h. Vertices at (1,−3) and (1, 1); asymptote y + 1 =3

2(x− 1)

Ans: −9(x− 1)2

16+

(y + 1)2

4= 1

8. Find the center, transverse axis, vertices, foci, and asymptotes. Graph theequation.

a.x2

25− y2

9= 1

Ans: Center: (0, 0); Transverse: y = 0; Vertices: (−5, 0), (5, 0);

Foci (−√34, 0), (

√34, 0); Asymptotes: y = ±3

5x

b. −x2

4+

y2

16= 1

Ans: Center: (0, 0); Transverse: x = 0; Vertices: (0,−4), (0, 4);Foci (0,−2

√5), (0, 2

√5); Asymptotes: y = ±2x

c. −x2 + 4y2 = 16

Ans: Center: (0, 0); Transverse: x = 0; Vertices: (0,−2), (0, 2);

Foci (0,−2√5), (0, 2

√5); Asymptotes: y = ±1

2x

d.(x− 2)2

4− (y + 3)2

9= 1

Ans: Center: (2,−3); Transverse: y = −3; Vertices: (0,−3), (4,−3);

Foci (2−√13,−3), (2 +

√13,−3); Asymptotes: y + 3 = ±3

2(x− 2)

e. −(x+ 2)2 +(y − 2)2

4= 1

Ans: Center: (−2, 2); Transverse: x = −2; Vertices: (−2, 0), (−2, 4);Foci (−2, 2−

√5), (−2, 2 +

√5); Asymptotes: y − 2 = ±2(x+ 2)

f.(x+ 1)2

4− (y + 2)2

4= 1

Ans: Center: (−1,−2); Transverse: y = −2; Vertices: (−3,−2), (1,−2);Foci (−1− 2

√2,−2), (−1 + 2

√2,−2); Asymptotes: y + 2 = ±(x+ 1)

g. x2 − y2 − 2x− 2y − 1 = 0

Ans: Center: (1,−1); Transverse: y = −1; Vertices: (0,−1), (2,−1);Foci (1−

√2,−1), (1 +

√2,−1); Asymptotes: y + 1 = ±(x− 1)

h. −4x2 − 8x+ y2 − 4y − 4 = 0

Ans: Center: (−1, 2); Transverse: x = −1; Vertices: (−1, 0), (−1, 4);Foci (−1, 2−

√5), (−1, 2 +

√5); Asymptotes: y − 2 = ±2(x+ 1)

i. 2x2 + 4x− y2 + 4y − 4 = 0

Ans: Center: (−1, 2); Transverse: y = 2; Vertices: (−2, 2), (0, 2);Foci (−1−

√3, 2), (−1 +

√3, 2); Asymptotes: y − 2 = ±

√2(x+ 1)

j. x2 + 8x− 3y2 − 6y + 4 = 0

Ans: Center: (−4,−1); Transverse: y = −1; Vertices: (−7,−1), (−1,−1);

Foci (−4− 2√3,−1), (−4 + 2

√3,−1); Asymptotes: y + 1 = ±

√3

3(x+ 4)