Algebra II
1 of 30
Conics 6 – Parabolas: SOLUTIONS
All Graphs in this packet were made using Mathematica.
All problems use the technique of completing the square on the general form of
the parabola to change it into the standard form of a parabola. The numbers
added when completing the square are shown in boxes.
1. x 2 + 6x − 12y + 57 = 0
(x
2
)
+ 6x + 9 = 12y − 57 + 9
10
( x + 3) = 12y − 48
( x + 3)2 = 12 ( y − 4 )
8
Vertex : ( −3, 4 )
6
4 p = 12 ⇒ p = 3
4
2
Focus : ( −3, 7 )
2
Directrix : y = 1
L : ( −9, 7 )
�10
�5
5
R : ( 3, 7 )
2. x 2 + 12x − 16y − 44 = 0
(x
2
)
+ 12x + 36 = 16y + 44 + 36
( x + 6 ) = 16y + 80
( x + 6 )2 = 16 ( y + 5 )
5
2
Vertex : ( −6, − 5 )
4 p = 16 ⇒ p = 4
Focus : ( −6, − 1)
�15
�10
�5
5
�5
Directrix : y = −9
L : ( −14, − 1)
�10
R : ( 2, − 1)
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
2 of 30
Conics 6 – Parabolas: SOLUTIONS
3. y 2 − 16x + 12y − 44 = 0
(y
2
)
+ 12x + 36 = 16x + 44 + 36
5
( y + 6 ) = 16x + 80
( y + 6 )2 = 16 ( x + 5 )
2
Vertex : ( −5, − 6 )
�10
�8
�6
�4
�2
2
4
4 p = 16 ⇒ p = 4
Focus : ( −1, − 6 )
�5
Directrix : x = −9
L : ( −1, 2 )
R : ( −1, − 14 )
�10
�15
4. x 2 + 6x − 8y + 65 = 0
(
)
15
x 2 + 6x + 9 = 8y − 65 + 9
( x + 3)2 = 8y − 56
( x + 3)2 = 8 ( y − 7 )
10
Vertex : ( −3, 7 )
4p = 8 ⇒ p = 2
Focus : ( −3, 9 )
5
Directrix : y = 5
L : ( −7, 9 )
R : ( −3, 9 )
�10
© 2010 Tim Howarth
�5
5
www.TimTheMathMan.com
Algebra II
3 of 30
Conics 6 – Parabolas: SOLUTIONS
5. x 2 + 8x − 4y + 28 = 0
(x
2
)
+ 8x + 16 = 4y − 28 + 16
8
( x + 4 ) = 4y − 12
( x + 4 )2 = 4 ( y − 3)
6
4p = 4 ⇒ p =1
4
Directrix : y = 2
2
2
Vertex : ( −4, 3)
Focus : ( −4, 4 )
L : ( −6, 4 )
R : ( −2, 4 )
�8
�6
�4
�2
2
6. y 2 + 16x − 8y + 64 = 0
(y
2
)
− 8y + 16 = −16x − 64 + 16
15
( y − 4 )2 = −16x − 48
( y − 4 )2 = −16 ( x + 3)
10
Vertex : ( −3, 4 )
4 p = 16 ⇒ p = 4
Focus : ( 7, 4 )
5
Directrix : x = 1
L : ( 7, 12 )
R : ( 7, − 4 )
�12
�10
�8
�6
�4
�2
2
�5
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
4 of 30
Conics 6 – Parabolas: SOLUTIONS
7. y 2 − 12x − 8y + 100 = 0
(y
2
)
− 8y + 16 = 12x − 100 + 16
( y − 4 )2 = 12x − 84
( y − 4 )2 = 12 ( x − 7 )
10
Vertex : ( 7, 4 )
4 p = 12 ⇒ p = 3
5
Focus : (10, 4 )
Directrix : x = 4
L : (10, 10 )
5
10
15
R : (10 − 2 )
�5
8. y 2 + 12x − 10y + 61 = 0
(y
2
)
15
− 10y + 25 = −12x − 61 + 25
( y − 5 )2 = −12x − 36
( y − 5 )2 = −12 ( x + 3)
10
Vertex : ( −3, 5 )
4 p = 12 ⇒ p = 3
Focus : ( −6, 5 )
5
Directrix : x = 0
L : ( −6, 11)
R : ( −6, − 1)
�12
�10
�8
�6
�4
�2
2
�5
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
5 of 30
Conics 6 – Parabolas: SOLUTIONS
9. x 2 + 14x + 12y + 133 = 0
(x
2
)
+ 14x + 49 = −12y − 133 + 49
�15
�10
�5
( x + 7 )2 = −12y − 84
( x + 7 )2 = −12 ( y + 7 )
�5
Vertex : ( −7, − 7 )
4 p = 12 ⇒ p = 3
Focus : ( −7, − 10 )
�10
Directrix : y = −4
L : ( −13, − 10 )
R : ( −1, − 10 )
�15
5
10. y 2 + 8x + 6y − 15 = 0
(y
2
)
+ 6y + 9 = −8x + 15 + 9
( y + 3)2 = −8x + 24
( y + 3)2 = −8 ( x − 3)
Vertex : ( 3, − 3)
�4
�2
2
4
6
4p = 8 ⇒ p = 2
Focus : (1, − 3)
Directrix : x = 5
L : (1, 1)
�5
R : (1, − 7 )
�10
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
6 of 30
Conics 6 – Parabolas: SOLUTIONS
11. x 2 − 14x − 12y + 133 = 0
(x
2
)
− 14x + 49 = 12y − 133 + 49
( x − 7 )2 = 12y − 84
( x − 7 )2 = 12 ( y − 7 )
15
Vertex : ( 7, 7 )
4 p = 12 ⇒ p = 3
Focus : ( 7, 10 )
10
Directrix : y = 4
L : (1, 10 )
R : (13, 10 )
5
5
10
15
12. y 2 + 4x − 8y + 16 = 0
(y
2
)
− 8y + 16 = −4x − 16 + 16
10
( y − 4 ) = −4x + 0
( y − 4 )2 = −4 ( x + 0 )
2
Vertex : ( 0, 4 )
8
4p = 4 ⇒ p =1
Focus : ( −1, 4 )
6
Directrix : x = 1
L : ( −1, 6 )
R : ( −1, 2 )
4
2
�4
© 2010 Tim Howarth
�2
2
www.TimTheMathMan.com
Algebra II
7 of 30
Conics 6 – Parabolas: SOLUTIONS
13. x 2 + 8x + 12y − 20 = 0
(x
2
)
+ 8x + 16 = −12y + 20 + 16
( x + 4 )2 = −12y + 36
( x + 4 )2 = −12 ( y − 3)
8
Vertex : ( −4, 3)
6
4 p = 12 ⇒ p = 3
4
Focus : ( −4, 0 )
2
Directrix : y = 6
L : ( −10, 0 )
�10
�5
R : ( 2, 0 )
5
�2
�4
14. y 2 + 16x − 12y + 100 = 0
(
)
20
y 2 − 12y + 36 = −16x − 100 + 36
( y − 6 )2 = −16x − 64
( y − 6 )2 = −16 ( x + 4 )
15
Vertex : ( −4, 6 )
4 p = 16 ⇒ p = 4
Focus : ( −8, 6 )
10
Directrix : x = 0
L : ( −8, 14 )
R : ( −8, − 2 )
5
�15
�10
�5
�5
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
8 of 30
Conics 6 – Parabolas: SOLUTIONS
15. x 2 − 14x − 12y + 97 = 0
(x
2
)
− 14x + 49 = 12y − 97 + 49
( x − 7 )2 = 12y − 48
( x − 7 )2 = 12 ( y − 4 )
Vertex : ( 7, 4 )
10
4 p = 12 ⇒ p = 3
8
Focus : ( 7, 7 )
6
Directrix : y = 1
L : (1, 7 )
4
R : (13, 7 )
2
5
10
15
16. x 2 + 8x + 8y − 40 = 0
(x
2
)
+ 8x + 16 = −8y + 40 + 16
( x + 4 )2 = −8y + 56
( x + 4 )2 = −8 ( y − 7 )
Vertex : ( −4, 7 )
10
4p = 8 ⇒ p = 2
Focus : ( −4, 7 )
8
Directrix : y = 9
L : ( −8, 7 )
6
R : ( 0, 7 )
4
2
�10
© 2010 Tim Howarth
�5
www.TimTheMathMan.com
5
Algebra II
9 of 30
Conics 6 – Parabolas: SOLUTIONS
17. y 2 − 20x − 14y − 11 = 0
(y
2
)
− 14y + 49 = 20x + 11 + 49
20
( y − 7 ) = 20x + 60
( y − 7 )2 = 20 ( x + 3)
2
Vertex : ( −3, 7 )
15
4 p = 20 ⇒ p = 5
Focus : ( 2, 7 )
10
Directrix : x = −8
L : ( 2, 17 )
5
R : ( 2, − 3)
�10
�5
5
10
�5
18. x 2 − 14x − 28y + 245 = 0
(x
2
)
− 14x + 49 = 28y − 245 + 49
( x − 7 )2 = 28y − 196
( x − 7 )2 = 28 ( y − 7 )
Vertex : ( 7, 7 )
4 p = 28 ⇒ p = 7
20
Focus : ( 7, 14 )
Directrix : y = 0
15
L : ( −7, 14 )
R : ( 21, 14 )
10
5
�10
© 2010 Tim Howarth
10
20
www.TimTheMathMan.com
30
Algebra II
10 of 30
Conics 6 – Parabolas: SOLUTIONS
19. y 2 + 16x + 6y − 119 = 0
(
)
10
y 2 + 6y + 9 = −16x + 119 + 9
( y + 3)2 = −16x + 128
( y + 3)2 = −16 ( x + 8 )
5
Vertex : ( −8, − 3)
4 p = 16 ⇒ p = 4
�5
Focus : ( 4, − 3)
5
10
Directrix : x = 12
L : ( 4, 5 )
�5
R : ( 4, − 11)
�10
�15
20. x 2 + 14x − 8y + 73 = 0
(x
2
)
+ 14x + 49 = 8y − 73 + 49
( x + 7 )2 = 8y − 24
( x + 7 )2 = 8 ( y − 3)
10
Vertex : ( −7, 3)
8
4p = 8 ⇒ p = 2
Focus : ( −7, 5 )
6
Directrix : y = 1
L : ( −11, 5 )
4
R : ( −3, 5 )
2
�15
�10
�5
�2
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
11 of 30
Conics 6 – Parabolas: SOLUTIONS
21. y 2 + 8x + 6y − 15 = 0
(y
2
)
+ 6y + 9 = −8x + 15 + 9
5
( y + 3)2 = −8x + 24
( y + 3)2 = −8 ( x − 3)
Vertex : ( 3, − 3)
4p = 8 ⇒ p = 2
�6
�4
�2
2
4
6
Focus : (1, − 3)
Directrix : x = 5
L : (1, 1)
�5
R : (1, − 7 )
�10
22. y 2 − 16x + 8y = 0
(y
2
)
+ 8y + 16 = 16x + 16
10
( y + 4 )2 = 16x + 16
( y + 4 )2 = 16 ( x + 1)
5
Vertex : ( −1, − 4 )
4 p = 16 ⇒ p = 4
Focus : ( −3, − 4 )
�5
5
10
Directrix : x = −5
L : ( 3, 4 )
R : ( 3, − 12 )
�5
�10
�15
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
12 of 30
Conics 6 – Parabolas: SOLUTIONS
23. y 2 + 4x − 8y = 0
(
)
10
y 2 − 8y + 16 = −4x + 16
( y − 4 )2 = −4x + 16
( y − 4 )2 = −4 ( x − 4 )
8
Vertex : ( 4, 4 )
6
4p = 4 ⇒ p =1
Focus : ( 3, 4 )
4
Directrix : x = 5
L : ( 3, 6 )
2
R : ( 3, 2 )
�4
�2
2
4
6
�2
24. x 2 + 10x − 20y + 65 = 0
(x
2
)
+ 10x + 25 = 20y − 65 + 25
( x + 5 )2 = 20y − 40
( x + 5 )2 = 20 ( y − 2 )
Vertex : ( −5, 2 )
4 p = 20 ⇒ p = 5
Focus : ( −5, 7 )
10
Directrix : y = −3
L : ( −15, 7 )
5
R : ( 5, 7 )
�20
�15
�10
�5
5
�5
© 2010 Tim Howarth
www.TimTheMathMan.com
10
Algebra II
13 of 30
Conics 6 – Parabolas: SOLUTIONS
25. x 2 + 6x + 8y + 9 = 0
(x
2
)
+ 6x + 9 = −8y − 9 + 9
( x + 3)2 = −8y
( x + 3)2 = −8 ( y + 0 )
4
Vertex : ( −3, 0 )
2
4p = 8 ⇒ p = 2
Focus : ( −3, − 2 )
�10
�5
5
Directrix : y = 2
�2
L : ( −7, − 2 )
R : (1, − 2 )
�4
�6
26. y 2 + 8x − 14y − 7 = 0
(y
2
)
− 14y + 49 = −8x + 7 + 49
( y − 7 )2 = −8x + 56
( y − 7 )2 = −8 ( x − 7 )
15
Vertex : ( 7, 7 )
4p = 8 ⇒ p = 2
Focus : ( 5, 7 )
Directrix : x = 9
10
L : ( 5, 11)
R : ( 5, 3)
5
�4
© 2010 Tim Howarth
�2
2
4
6
8
10
www.TimTheMathMan.com
Algebra II
14 of 30
Conics 6 – Parabolas: SOLUTIONS
27. x 2 + 6x + 12y + 57 = 0
(x
2
)
+ 6x + 9 = −12y − 57 + 9
( x + 3)2 = −12y − 48
( x + 3)2 = −12 ( y + 4 )
Vertex : ( −3, − 4 )
�10
�5
5
4 p = 12 ⇒ p = 3
�2
Focus : ( −3, − 7 )
�4
Directrix : y = −1
L : ( −9, − 7 )
�6
R : ( 3, − 7 )
�8
�10
28. x 2 + 6x + 20y + 89 = 0
(x
2
)
+ 6x + 9 = −20y − 89 + 9
( x + 3)2 = −20y − 80
( x + 3)2 = −20 ( y + 4 )
Vertex : ( −3, − 4 )
4 p = 20 ⇒ p = 5
Focus : ( −3, − 9 )
Directrix : y = 1
L : ( −13, − 7 )
R : ( 7, − 7 )
�20
�15
�10
�5
5
10
�5
�10
�15
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
15 of 30
Conics 6 – Parabolas: SOLUTIONS
29. x 2 − 16x + 8y = 0
(x
2
)
− 16x + 64 = −8y + 64
( x − 8 )2 = −8y + 64
( x − 8 )2 = −8 ( y − 8 )
Vertex : ( 8, 8 )
4p = 8 ⇒ p = 2
Focus : ( 8, 6 )
12
10
8
Directrix : y = 10
L : ( 4, 6 )
6
R : (12, 6 )
4
2
5
10
15
30. y 2 + 12x − 18y + 117 = 0
(y
2
)
− 18y + 81 = −12x − 117 + 81
( y − 9 )2 = −12x − 36
( y − 9 )2 = −12 ( x + 3)
20
4 p = 12 ⇒ p = 3
15
Vertex : ( −3, 9 )
Focus : ( −6, 9 )
Directrix : x = 0
L : ( −6, 15 )
10
R : ( −6, 3)
5
�10
© 2010 Tim Howarth
�5
www.TimTheMathMan.com
Algebra II
16 of 30
Conics 6 – Parabolas: SOLUTIONS
31. y 2 − x − 14y + 46 = 0
(y
2
)
− 14y + 49 = x − 46 + 49
8
( y − 7) = x + 3
2
( y − 7 )2 = 1( x + 3)
Vertex : ( −3, 7 )
4p =1⇒ p =
6
1
4
Focus : ( −2 43 , 7 )
Directrix : x = −3 14
4
L : ( −2 43 , 7 12 )
R : ( −2 43 , 6 12 )
2
�4
�3
�2
�1
1
1
32. y 2 − x + 4y + 1 = 0
(y
2
)
+ 4y + 4 = x − 1 + 4
( y + 2 )2 = x + 3
( y + 2 )2 = 1( x + 3)
Vertex : ( −3, − 2 )
4p =1⇒ p =
�4
�3
�2
�1
1
�1
1
4
Focus : ( −2 43 , − 2 )
Directrix : x = −3
L : ( −2 43 , − 1 12 )
�2
1
4
R : ( −2 43 , − 2 12 )
�3
�4
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
17 of 30
Conics 6 – Parabolas: SOLUTIONS
33. y 2 − 4x − 16y + 65 = 0
(
)
12
y 2 − 16y + 64 = 4x − 65 + 64
( y − 8 )2 = 4x − 1
( y − 8 )2 = 4 ( x − 14 )
10
Vertex : ( 14 , 8 )
8
4p = 4 ⇒ p =1
Focus : (1 14 , 8 )
6
Directrix : x = − 43
L : (1 14 , 10 )
4
R : (1 14 , 6 )
2
�2
�1
1
2
3
4
34. x 2 + 6x − 6y + 21 = 0
(x
2
)
+ 6x + 9 = 6y − 21 + 9
( x + 3)2 = 6y − 12
( x + 3)2 = 6 ( y − 2 )
8
Vertex : ( −3, 2 )
4 p = 6 ⇒ p = 1 12
Focus : ( −3, 3
Directrix : y =
L : ( −6, 3 12 )
1
2
6
)
1
2
4
R : ( 0, 3 12 )
2
�8
© 2010 Tim Howarth
�6
�4
�2
2
www.TimTheMathMan.com
4
Algebra II
18 of 30
Conics 6 – Parabolas: SOLUTIONS
35. y 2 − 10x − 10y + 55 = 0
(y
2
)
− 10y + 25 = 10x − 55 + 25
( y − 5 )2 = 10x − 30
( y − 5 )2 = 10 ( x − 3)
Vertex : ( 3, 5 )
10
4 p = 10 ⇒ p = 2 12
Focus : ( 5 12 , 5 )
Directrix : x =
L : ( 5 12 , 10 )
1
2
5
R : ( 5 12 , 0 )
�2
2
4
6
8
10
36. x 2 + 7x − 10y − 27 43 = 0
(x
2
)
+ 7x + 12 14 = 10y + 27 43 + 12 14
( x + 3 ) = 10y + 40
( x + 3 ) = 10 ( y + 4 )
Vertex : ( −3 , − 4 )
1 2
2
1 2
2
4
1
2
4 p = 10 ⇒ p = 2 12
2
Focus : ( −3 12 , − 1 12 )
Directrix : y = −6 12
L : ( −8 12 , − 1 12 )
R : (1 , − 1
1
2
1
2
�10
�5
5
�2
)
�4
�6
�8
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
19 of 30
Conics 6 – Parabolas: SOLUTIONS
37. x 2 + 6x + 10y − 31 = 0
(x
2
)
+ 6x + 9 = −10y + 31 + 9
( x + 3)2 = −10y + 40
( x + 3)2 = −10 ( y − 4 )
8
Vertex : ( −3, 4 )
6
4 p = 10 ⇒ p = 2 12
4
Focus : ( −3, 1 12 )
Directrix : y = 6 12
2
L : ( −8, 1 12 )
R : ( 2, 1 12 )
�10
�5
5
�2
�4
38. x 2 − 12x + 6y = 0
(x
2
)
− 12x + 36 = −6y + 36
( x − 6 )2 = −6y + 36
( x − 6 )2 = −6 ( y − 6 )
8
Vertex : ( 6, 6 )
4 p = 6 ⇒ p = 1 12
6
Directrix : y = 7 12
4
R : ( 9, 4 12 )
2
Focus : ( 6, 4 12 )
L : ( 3, 4 12 )
5
© 2010 Tim Howarth
10
www.TimTheMathMan.com
Algebra II
20 of 30
Conics 6 – Parabolas: SOLUTIONS
39. y 2 + 14x + 7y − 78.75 = 0
(y
2
)
+ 7y + 12.25 = −14x + 78.75 + 12.25
( y + 3.5 )2 = −14x + 91
( y + 3.5 )2 = −14 ( x − 6.5 )
5
Vertex : ( 6.5, − 3.5 )
4 p = 14 ⇒ p = 3.5
Focus : ( 3, − 3.5 )
�2
Directrix : x = 10
2
4
6
8
10
12
L : ( 3, 3.5 )
R : ( 3, − 10.5 )
�5
�10
40. x 2 − 6x − 13y + 105.2 = 0
(x
2
)
− 6x + 9 = 13y − 105.2 + 9
( x − 3)2 = 13y − 96.2
( x − 3)2 = 13( y − 7.4 )
Vertex : ( 3, 7.4 )
15
4 p = 13 ⇒ p = 3.25
Focus : ( −3, 10.65 )
Directrix : y = 4.15
10
L : ( −9.5, 10.65 )
R : ( 3.5, 10.65 )
5
�15
© 2010 Tim Howarth
�10
�5
5
www.TimTheMathMan.com
Algebra II
21 of 30
Conics 6 – Parabolas: SOLUTIONS
41. y 2 + 15.2x − 4.2y − 103.51 = 0
(y
2
)
− 4.2y + 4.41 = −15.2x + 103.51 + 4.41
( y − 2.1)2 = −15.2x + 107.92
( y − 2.1)2 = −15.2 ( x − 7.1)
10
Vertex : ( 7.1, 2.1)
5
4 p = 15.2 ⇒ p = 3.8
Focus : ( 3.3, 2.1)
Directrix : x = 10.9
L : ( 3.3, 9.7 )
�2
2
4
6
8
10
12
R : ( 3.3, − 5.5 )
�5
�10
42. x 2 + 8x − 15y + 121 = 0
(x
2
)
+ 8x + 16 = 15y − 121 + 16
( x + 4 )2 = 15y − 105
( x + 4 )2 = 15 ( y − 7 )
Vertex : ( −4, 7 )
4 p = 15 ⇒ p = 3.75
15
Focus : ( −4, 10.75 )
Directrix : y = 3.25
L : ( −11.5, 10.75 )
10
R : ( 3.5, 10.75 )
5
�15
© 2010 Tim Howarth
�10
�5
5
www.TimTheMathMan.com
Algebra II
22 of 30
Conics 6 – Parabolas: SOLUTIONS
43. x 2 − 9.4x + 12.25y − 86.5675 = 0
(x
2
)
− 9.4x + 22.09 = −12.25y + 86.5675 + 22.09
( x − 4.7 )2 = −12.25y + 108.6575
( x − 4.7 )2 = −12.25 ( y − 8.87 )
14
Vertex : ( 4.7, 8.87 )
12
4 p = 12.25 ⇒ p = 3.0625
10
Focus : ( 4.7, 5.8075 )
Directrix : y = 11.9325
8
L : ( −1.425, 5.8075 )
6
R : (10.825, 5.8075 )
4
2
�5
5
10
15
44. y 2 + 27x + 15.6y − 101.16 = 0
(y
2
)
+ 15.6y + 60.84 = −27x + 101.16 + 60.84
( y + 7.8 )2 = −27x + 162
( y + 7.8 )2 = −27 ( x − 6 )
10
Vertex : ( 6, − 7.8 )
4 p = 27 ⇒ p = 6.75
Focus : ( −0.75, − 7.8 )
�5
5
10
Directrix : x = 12.75
L : ( −0.75, − 21.3)
R : ( −0.75, 5.7 )
�10
�20
© 2010 Tim Howarth
www.TimTheMathMan.com
15
Algebra II
23 of 30
Conics 6 – Parabolas: SOLUTIONS
45. y 2 − 12.5x − 13.6y + 136.24 = 0
(y
2
)
− 13.6y + 46.24 = 12.5x − 136.24 + 46.24
( y − 6.8 )2 = 12.5x − 90
( y − 6.8 )2 = 12.5 ( x − 7.2 )
15
Vertex : ( 7.2, 6.8 )
4 p = 12.5 ⇒ p = 3.125
Focus : (10.325, 6.8 )
10
Directrix : x = 4.075
L : (10.325, 13.05 )
R : (10.325, 0.95 )
5
5
10
15
46. x 2 + 10x + 10.28y + 68.176 = 0
(x
2
)
+ 10x + 25 = −10.28y − 68.176 + 25
( x + 5 )2 = −10.28y − 43.176
( x + 5 )2 = −10.28 ( y + 4.2 )
Vertex : ( −5, − 4.2 )
4 p = 10.28 ⇒ p = 2.57
Focus : ( −5, − 6.77 )
Directrix : y = −1.63
L : ( −10.14, − 6.77 )
R : ( 0.14, − 6.77 )
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5
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© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
24 of 30
Conics 6 – Parabolas: SOLUTIONS
143
47. x 2 + 6 49 x − 12y + 44 162
=0
(x
2
)
31
143
31
+ 6 49 x + 10 81
= 12y − 44 162
+ 10 81
( x + 3 ) = 12y − 34
( x + 3 ) = 12 ( y − 2 )
Vertex : ( −3 , 2 )
2 2
9
1
2
2 2
9
7
8
2
9
10
7
8
8
4 p = 12 ⇒ p = 3
Focus : ( −3 29 , 5 87 )
6
Directrix : y = − 18
L : ( −9 29 , 5 87 )
4
R : ( 2 97 , 5 87 )
2
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5
�2
47
48. y 2 − 14x − 1 43 y − 100 64
=0
(y
2
− 1 43 y +
(y − )
(y − )
) = 14x + 100
49
64
7 2
8
= 14x + 101 12
7 2
8
= 14 ( x + 7 14 )
Vertex : ( −7 14 ,
7
8
47
64
+
49
64
15
)
10
4 p = 14 ⇒ p = 3 12
Focus : ( −3 43 ,
7
8
)
5
Directrix : x = −10 43
L : ( −3 43 , 7 87 )
R : ( −3 , − 6
3
4
1
8
)
�10
�5
5
�5
�10
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
25 of 30
Conics 6 – Parabolas: SOLUTIONS
49. y 2 + 19x + 5 13 y − 99 55
72 = 0
(y
2
)
1
+ 5 13 y + 7 19 = −19x + 99 55
72 + 7 9
10
( y + 2 ) = −19x + 106
( y + 2 ) = −19 ( x − 5 )
Vertex : ( −2 , 5 )
2 2
3
7
8
2 2
3
5
8
2
3
5
5
8
4 p = 19 ⇒ p = 4 43
Focus : ( 87 , − 2 23 )
�5
Directrix : x = 10 83
5
10
�5
L : ( 87 , 6 56 )
R : ( 87 , − 12 16 )
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41
50. x 2 − 8 49 x − 12y − 15 81
=0
(x
2
)
41
− 8 49 x + 17 67
= 12y + 15 81
+ 17 67
81
81
( x − 4 ) = 12y + 33
( x − 4 ) = 12 ( y + 2 )
Vertex : ( 4 , − 2 )
2 2
9
1
3
2 2
9
7
9
2
9
7
9
6
4 p = 12 ⇒ p = 3
Focus : ( 4 29 ,
2
9
)
4
Directrix : y = −5 97
L : ( −1 ,
7
9
R : (10 29 ,
2
9
2
9
)
)
2
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5
10
�2
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�8
© 2010 Tim Howarth
www.TimTheMathMan.com
15
Algebra II
26 of 30
Conics 6 – Parabolas: SOLUTIONS
51. x 2 + 14 47 x + 16y + 46 167
245 = 0
(x
2
)
4
+ 14 47 x + 53 494 = −16y − 46 167
245 + 53 49
( x + 7 ) = −16y + 6
( x + 7 ) = −16 ( y − )
Vertex : ( −7 , )
2 2
7
2
5
2 2
7
2
5
2
7
6
2
5
4
4 p = 16 ⇒ p = 4
Focus : ( −7 27 , − 3 53 )
2
Directrix : y = 4 25
L : ( −15 , − 3
2
7
R :( , − 3
5
7
3
5
)
3
5
�15
)
�10
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5
�2
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68
52. x 2 − 6 47 x − 12 29 y + 67 147
=0
(x
2
)
68
− 6 47 x + 10 39
= 12 29 y − 67 147
+ 10 39
49
49
( x − 3 ) = 12y − 56
( x − 3 ) = 12 ( y − 4 )
Vertex : ( 3 , 4 )
2 2
7
2
3
2 2
7
7
11
2
7
7
11
12
4 p = 12 ⇒ p = 3
Focus : ( 3 27 , 7 137
198 )
Directrix : y =
L : ( −2 , 7
52
63
137
198
137
R : ( 9 25
63 , 7 198 )
10
29
126
8
)
6
4
2
�5
© 2010 Tim Howarth
5
10
www.TimTheMathMan.com
Algebra II
27 of 30
Conics 6 – Parabolas: SOLUTIONS
53. y 2 − 21 132 x − 4y + 152 131 = 0
(y
2
)
− 4y + 4 = 21 132 x − 152 131 + 4
( y − 2 )2 = 21 132 x − 148 131
( y − 2 )2 = 21 132 ( x − 7 )
15
10
Vertex : ( 7, 2 )
4 p = 21 132 ⇒ p = 5 15
52
5
Focus : (12 15
52 , 2 )
37
Directrix : x = 1 52
L : (12 , 12
15
52
15
26
5
)
15
R : (12 15
52 , − 8 26 )
10
15
20
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�10
�15
72
54. y 2 + 24 67 x + 15 53 y + 309 175
=0
(y
2
)
21
72
21
+ 15 53 y + 60 25
= −24 67 x − 309 175
+ 60 25
( y + 7 ) = −24 x − 248
( y + 7 ) = −24 ( x + 10 )
Vertex : ( −10, − 7 )
4 2
5
6
7
4 2
5
6
7
4
7
10
4
5
4 p = 24 67 ⇒ p = 6 143
Focus : ( −16 143 , − 7 45 )
�25
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�15
�10
�5
11
Directrix : x = −3 14
L : ( −16 143 , 4 22
35 )
R : ( −16 143 , − 20 358 )
�10
�20
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
28 of 30
Conics 6 – Parabolas: SOLUTIONS
55. x 2 − 6 47 x + 16y − 111 128
147 = 0
(x
2
)
39
− 6 47 x + 10 39
= −16y + 111 128
49
147 + 10 49
( x − 3 ) = −16y + 122
( x − 3 ) = −16 ( y − 7 )
Vertex : ( 3 , 7 )
2 2
7
2
3
2 2
7
14
2
3
2
7
2
3
12
4 p = 16 ⇒ p = 4
Focus : ( 3 , 3
2
7
2
3
10
)
8
Directrix : y = 11 23
L : ( −4 57 , 3 23 )
6
R : (11 27 , 3 23 )
4
2
�5
56. Note where each axis begins on
the first graph. The second graph is
if each axis was normal.
y 2 − 14 x + 13 13 y + 46 367 = 0
(y
2
)
+ 13 13 y + 44 49 = 14 x − 46 367 + 44 49
(y + 6 ) = x − 1
( y + 6 ) = ( x − 7)
Vertex : ( 7, − 6 )
2 2
3
1
4
2 2
3
1
4
3
4
2
3
4p = ⇒ p =
1
4
5
10
15
�6.2
�6.4
�6.6
�6.8
�7.0
6.6
6.8
7.0
7.2
7.4
2
4
6
8
1
16
Focus : ( 7 161 , − 6 23 )
Directrix : x = 6 15
16
L : ( 7 161 , − 6 13
24 )
R :(7 , − 6
1
16
19
24
)
�2
�4
�6
�8
© 2010 Tim Howarth
www.TimTheMathMan.com
Algebra II
29 of 30
Conics 6 – Parabolas: SOLUTIONS
20
57. y 2 + 14x − 18 12 y + 89 169 = 0
(y
2
)
− 18 12 y + 85 169 = −14x − 89 169 + 85 169
( y − 9 ) = −14x − 4
( y − 9 ) = −14 ( x + )
Vertex : ( − , 9 )
1 2
4
1 2
4
15
2
7
2
7
1
4
4 p = 14 ⇒ p = 3 12
11
Focus : ( −3 14
, 9 14 )
10
Directrix : x = 3 143
11
L : ( −3 14
, 16 14 )
11
R : ( −3 14
, 2 14 )
5
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2
4
58. x 2 + 8 25 x + 17 73 y − 23 752 = 0
(x
2
)
+ 8 25 x + 17 16
= −17 73 y + 23 752 + 17 16
25
25
( x + 4 ) = −17 y + 40
( x + 4 ) = −17 ( y − 2 )
Vertex : ( −4 , 2 )
1 2
5
3
7
1 2
5
3
7
1
5
2
3
1
3
1
3
4 p = 17 73 ⇒ p = 4 145
Focus : ( −4 15 , − 2 421 )
5
Directrix : y = 6 29
42
1
L : ( −12 32
35 , − 2 42 )
1
R : ( 4 18
35 , − 2 42 )
�15
�10
�5
5
�5
© 2010 Tim Howarth
www.TimTheMathMan.com
10
Algebra II
30 of 30
Conics 6 – Parabolas: SOLUTIONS
4
1888
59. x 2 − 14 14
19 x + 13 5 y + 104 3971 = 0
(x
2
)
106
106
− 14 14
= −13 45 y − 104 1888
19 x + 54 361
3971 + 54 361
( x − 7 ) = −13 y − 50
( x − 7 ) = −13 ( y + 3 )
Vertex : ( 7 , − 3 )
7 2
19
4
5
7 2
19
4
5
7
19
2
11
7
11
2
7
11
4 p = 13 45 ⇒ p = 3 209
�5
5
19
Focus : ( 7 197 , − 7 220
)
�2
41
Directrix : y = − 220
�4
89
19
L : ( 190
, − 7 220
)
10
15
�6
51
19
R : (14 190
, − 7 220
)
�8
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697
60. y 2 + 25 18 x + 10 57 y − 74 1176
=0
(y
2
)
697
+ 10 57 y + 28 137
= −25 18 x + 74 1176
+ 28 137
196
196
( y + 5 ) = −25 x + 103
( y + 5 ) = −25 ( x − 4 )
Vertex : ( 4 , − 5 )
5 2
14
1
8
5 2
14
1
8
1
9
49
Focus : ( −2 288
, − 5 145 )
Directrix : x = 10 113
288
L : ( −2
,7
1
9
5
5
14
4 p = 25 18 ⇒ p = 6 329
49
288
10
7
24
23
112
)
49
103
R : ( −2 288
, − 17 112
)
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5
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© 2010 Tim Howarth
www.TimTheMathMan.com
10