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1 Algebra II 1 of 30 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 + x 1y + 7 = 0 ( x + x + 9 ) = 1y ( x + 3) = 1y ( x + 3) = 1( y ) Vertex : 3, p = 1 p = 3 Focus : 3, 7 Directrix : y = 1 L : 9, 7 R : 3, 7. x + 1x 1y = 0 ( x + 1x + 3 ) = 1y ( x + ) = 1y + 0 ( x + ) = 1( y + ) Vertex :, p = 1 p = Focus :, 1 Directrix : y = 9 L : 1, 1 R :, Tim Howarth
2 Algebra II of y 1x + 1y = 0 ( y + 1x + 3 ) = 1x ( y + ) = 1x + 0 ( y + ) = 1( x + ) Vertex :, p = 1 p = Focus : 1, Directrix : x = 9 L : 1, R : 1, 1 1. x + x y + = 0 ( x + x + 9 ) = y + 9 ( x + 3) = y ( x + 3) = ( y 7) Vertex : 3, 7 p = p = Focus : 3, 9 Directrix : y = L : 7, 9 R : 3, Tim Howarth
3 Algebra II 3 of 30. x + x y + = 0 ( x + x + 1 ) = y + 1 ( x + ) = y 1 ( x + ) = ( y 3) Vertex :, 3 p = p = 1 Focus :, Directrix : y = L :, R :,. y + 1x y + = 0 ( y y + 1 ) = 1x + 1 ( y ) = 1x ( y ) = 1( x + 3) Vertex : 3, p = 1 p = Focus : 7, Directrix : x = 1 L : 7, 1 R : 7, Tim Howarth
4 Algebra II of y 1x y + 0 = 0 ( y y + 1 ) = 1x ( y ) = 1x ( y ) = 1( x 7) Vertex : 7, p = 1 p = 3 Focus :, Directrix : x = L :, R : 1. y + 1x y + 1 = 0 ( y y + ) = 1x 1+ ( y ) = 1x 3 ( y ) = 1( x + 3) Vertex : 3, p = 1 p = 3 Focus :, Directrix : x = 0 L :, 11 R :, Tim Howarth
5 Algebra II of x + 1x + 1y = 0 ( x + 1x + 9 ) = 1y ( x + 7) = 1y ( x + 7) = 1( y + 7) Vertex : 7, 7 p = 1 p = 3 Focus : 7, Directrix : y = L : 13, R : 1, 1 1. y + x + y 1 = 0 ( y + y + 9 ) = x ( y + 3) = x + ( y + 3) = ( x 3) Vertex : 3, 3 p = p = Focus : 1, 3 Directrix : x = L : 1, 1 R : 1, 7 0 Tim Howarth
6 Algebra II of x 1x 1y = 0 ( x 1x + 9 ) = 1y ( x 7) = 1y ( x 7) = 1( y 7) Vertex : 7, 7 p = 1 p = 3 Focus : 7, Directrix : y = L : 1, R : 13, y + x y + 1 = 0 ( y y + 1 ) = x ( y ) = x + 0 ( y ) = ( x + 0) Vertex : 0, p = p = 1 Focus : 1, Directrix : x = 1 L : 1, R : 1, 0 Tim Howarth
7 Algebra II 7 of x + x + 1y 0 = 0 ( x + x + 1 ) = 1y ( x + ) = 1y + 3 ( x + ) = 1( y 3) Vertex :, 3 p = 1 p = 3 Focus :, 0 Directrix : y = L :, 0 R :, 0 1. y + 1x 1y + 0 = 0 ( y 1y + 3 ) = 1x ( y ) = 1x ( y ) = 1( x + ) Vertex :, p = 1 p = Focus :, Directrix : x = 0 L :, 1 R :, Tim Howarth
8 Algebra II of x 1x 1y + 97 = 0 ( x 1x + 9 ) = 1y ( x 7) = 1y ( x 7) = 1( y ) Vertex : 7, p = 1 p = 3 Focus : 7, 7 Directrix : y = 1 L : 1, 7 R : 13, x + x + y 0 = 0 ( x + x + 1 ) = y ( x + ) = y + ( x + ) = ( y 7) Vertex :, 7 p = p = Focus :, 7 Directrix : y = 9 L :, 7 R : 0, 7 0 Tim Howarth
9 Algebra II 9 of y 0x 1y 11 = 0 ( y 1y + 9 ) = 0x ( y 7) = 0x + 0 ( y 7) = 0( x + 3) Vertex : 3, 7 p = 0 p = Focus :, 7 Directrix : x = L :, 17 R :, x 1x y + = 0 ( x 1x + 9 ) = y + 9 ( x 7) = y 19 ( x 7) = ( y 7) Vertex : 7, 7 p = p = 7 Focus : 7, 1 Directrix : y = 0 L : 7, 1 R : 1, Tim Howarth
10 Algebra II of y + 1x + y 119 = 0 ( y + y + 9 ) = 1x ( y + 3) = 1x + 1 ( y + 3) = 1( x + ) Vertex :, 3 p = 1 p = Focus :, 3 Directrix : x = 1 L :, R :, x + 1x y + 73 = 0 ( x + 1x + 9 ) = y ( x + 7) = y ( x + 7) = ( y 3) Vertex : 7, 3 p = p = Focus : 7, Directrix : y = 1 L : 11, R : 3, 1 0 Tim Howarth
11 Algebra II 11 of y + x + y 1 = 0 ( y + y + 9 ) = x ( y + 3) = x + ( y + 3) = ( x 3) Vertex : 3, 3 p = p = Focus : 1, 3 Directrix : x = L : 1, 1 R : 1, 7. y 1x + y = 0 ( y + y + 1 ) = 1x + 1 ( y + ) = 1x + 1 ( y + ) = 1( x + 1) Vertex : 1, p = 1 p = Focus : 3, Directrix : x = L : 3, R : 3, Tim Howarth
12 Algebra II 1 of y + x y = 0 ( y y + 1 ) = x + 1 ( y ) = x + 1 ( y ) = ( x ) Vertex :, p = p = 1 Focus : 3, Directrix : x = L : 3, R : 3,. x + x 0y + = 0 ( x + x + ) = 0y + ( x + ) = 0y 0 ( x + ) = 0( y ) Vertex :, p = 0 p = Focus :, 7 Directrix : y = 3 L : 1, 7 R :, Tim Howarth
13 Algebra II 13 of 30. x + x + y + 9 = 0 ( x + x + 9 ) = y ( x + 3) = y ( x + 3) = ( y + 0) Vertex : 3, 0 p = p = Focus : 3, Directrix : y = L : 7, R : 1,. y + x 1y 7 = 0 ( y 1y + 9 ) = x ( y 7) = x + ( y 7) = ( x 7) Vertex : 7, 7 p = p = Focus :, 7 Directrix : x = 9 L :, 11 R :, Tim Howarth
14 Algebra II 1 of x + x + 1y + 7 = 0 ( x + x + 9 ) = 1y ( x + 3) = 1y ( x + 3) = 1( y + ) Vertex : 3, p = 1 p = 3 Focus : 3, 7 Directrix : y = 1 L : 9, 7 R : 3, 7. x + x + 0y + 9 = 0 ( x + x + 9 ) = 0y ( x + 3) = 0y 0 ( x + 3) = 0( y + ) Vertex : 3, p = 0 p = Focus : 3, 9 Directrix : y = 1 L : 13, 7 R : 7, Tim Howarth
15 Algebra II 1 of x 1x + y = 0 ( x 1x + ) = y + ( x ) = y + ( x ) 1 = ( y ) Vertex :, p = p = Focus :, Directrix : y = L :, R : 1, y + 1x 1y = 0 ( y 1y + 1 ) = 1x ( y 9) = 1x 3 ( y 9) = 1( x + 3) Vertex : 3, 9 p = 1 p = 3 Focus :, 9 Directrix : x = 0 L :, 1 R :, Tim Howarth
16 Algebra II 1 of y x 1y + = 0 ( y 1y + 9 ) = x + 9 ( y 7) = x + 3 ( y 7) = 1( x + 3) Vertex : 3, 7 p = 1 p = 1 Focus : 3, 7 Directrix : x = 3 1 L : 3, 7 1 R : 3, y x + y + 1 = 0 ( y + y + ) = x 1 + ( y + ) = x + 3 ( y + ) = 1( x + 3) Vertex : 3, p = 1 p = 1 Focus : 3, Directrix : x = 3 1 L : 3, 1 1 R : 3, Tim Howarth
17 Algebra II 17 of y x 1y + = 0 ( y 1y + ) = x + ( y ) = x 1 ( y ) = ( x 1 ) 1 Vertex :, p = p = 1 Focus : 1 1, Directrix : x = 3 L : 1 1, R : 1 1, x + x y + 1 = 0 ( x + x + 9 ) = y ( x + 3) = y 1 ( x + 3) = ( y ) Vertex : 3, p = p = 1 1 Focus : 3, 3 1 Directrix : y = 1 L :, 3 1 R : 0, Tim Howarth
18 Algebra II 1 of y x y + = 0 ( y y + ) = x + ( y ) = x 30 ( y ) = ( x 3) Vertex : 3, p = p = 1 Focus : 1, Directrix : x = 1 L : 1, R : 1, 0 3. x + 7x y 7 3 = 0 ( x + 7x ) = y ( x ) = y + 0 ( x ) = y + Vertex : ( 3 1, ) p = p = 1 Focus : 3 1, 1 1 Directrix : y = 1 L : 1, 1 1 R : 1 1, Tim Howarth
19 Algebra II 19 of x + x + y 31 = 0 ( x + x + 9 ) = y ( x + 3) = y + 0 ( x + 3) = ( y ) Vertex : 3, p = p = 1 Focus : 3, 1 1 Directrix : y = 1 L :, 1 1 R :, x 1x + y = 0 ( x 1x + 3 ) = y + 3 ( x ) = y + 3 ( x ) = ( y ) Vertex :, p = p = 1 1 Focus :, 1 Directrix : y = 7 1 L : 3, 1 R : 9, 1 0 Tim Howarth
20 Algebra II 0 of y + 1x + 7y 7.7 = 0 ( y + 7y + 1. ) = 1x ( y + 3.) = 1x + 91 ( y + 3.) = 1( x.) Vertex :., 3. p = 1 p = 3. Focus : 3, 3. Directrix : x = L : 3, 3. R : 3, x x 13y +. = 0 ( x x + 9 ) = 13y. + 9 ( x 3) = 13y 9. ( x 3) = 13( y 7.) Vertex : 3, 7. p = 13 p = 3. Focus : 3,. Directrix : y =.1 L : 9.,. R : 3., Tim Howarth
21 Algebra II 1 of y + 1.x.y 3.1 = 0 ( y.y +.1 ) = 1.x ( y.1) = 1.x ( y.1) = 1.( x 7.1) Vertex : 7.1,.1 p = 1. p = 3. Focus : 3.3,.1 Directrix : x =.9 L : 3.3, 9.7 R : 3.3,. 1. x + x 1y + 11 = 0 ( x + x + 1 ) = 1y ( x + ) = 1y ( x + ) = 1( y 7) Vertex :, 7 p = 1 p = 3.7 Focus :,.7 Directrix : y = 3. L : 11.,.7 R : 3., Tim Howarth
22 Algebra II of x 9.x + 1.y.7 = 0 ( x 9.x +.09 ) = 1.y ( x.7) = 1.y +.7 ( x.7) 1 = 1.( y.7) Vertex :.7,.7 p = 1. p = 3.0 Focus :.7,.07 Directrix : y = L : 1.,.07 R :., y + 7x + 1.y 1.1 = 0 ( y + 1.y + 0. ) = 7x ( y + 7.) = 7x + 1 ( y + 7.) = 7( x ) Vertex :, 7. p = 7 p =.7 Focus : 0.7, 7. Directrix : x = 1.7 L : 0.7, 1.3 R : 0.7, Tim Howarth
23 Algebra II 3 of 30. y 1.x 13.y = 0 ( y 13.y +. ) = 1.x ( y.) = 1.x 90 ( y.) 1 = 1.( x 7.) Vertex : 7.,. p = 1. p = 3.1 Focus :.3,. Directrix : x =.07 L :.3, 13.0 R :.3, x + x +.y +.17 = 0 ( x + x + ) =.y.17 + ( x + ) =.y 3.17 ( x + ) =.( y +.) Vertex :,. p =. p =.7 Focus :,.77 Directrix : y = 1.3 L :.1,.77 R : 0.1, Tim Howarth
24 Algebra II of x x 1y + = 0 1 x + 31 ( 9 x + 1 ) = 1y ( x ) = 1y 3 1 ( x ) = 1 y 7 Vertex : 3, 7 9 p = 1 p = 3 Focus : 3 9, 7 Directrix : y = 1 L : 9 9, 7 R : 7 9, 7. y 1x y 0 = 0 ( y 1 3 y + 9 ) = 1x ( y 7 ) = 1x ( y 7 ) = 1( x ) Vertex : 7 1, 7 p = 1 p = 3 1 Focus : 3 3, 7 Directrix : x = 3 L : 3 3, 7 7 R : 3 3, Tim Howarth
25 Algebra II of y + 19x y 99 = 0 7 ( y + 1 y ) = 19x ( y + 3) = 19x + 7 ( y + 3) = 19( x ) Vertex : 3, p = 19 p = 3 Focus : 7, 3 Directrix : x = 3 L : 7, R : 7, x 1 9 x 1y 1 = 0 1 x 7 ( 9 x ) = 1y ( x 9 ) = 1y ( x 9 ) = 1 y Vertex :, p = 1 p = 3 Focus : 9, 9 Directrix : y = L : 1 7 9, 9 R : 9, 9 0 Tim Howarth
26 Algebra II of x x + 1y + = 0 ( x + 1 x ) = 1y ( x ) = 1y + ( x ) = 1( y ) Vertex : 7, 7 p = 1 p = Focus : 7 7, 3 3 Directrix : y = L : 1 7, 3 3 R : 7, x x y + 7 = 0 17 x 39 ( 7 x + 9 ) = y ( x 3 7 ) = 1y 3 ( x 3 7 ) = 1( y 7 11) Vertex : 3 7, 7 11 p = 1 p = 3 Focus : 3 7, Directrix : y = 9 1 L :, R : 9, Tim Howarth
27 Algebra II 7 of y x y + 1 = 0 13 ( y y + ) = 1 x ( y ) = 1 x ( y ) = 1 ( x 7) 13 Vertex : 7, p = 1 13 p = 1 Focus : 1 1, Directrix : x = (, 1 ) L : 1 1 R : 1 1, y + x y = 0 17 y ( y + 0 ) = x ( y + 7 ) = x 7 7 ( y + 7 ) = 7 x + Vertex :, 7 p = 7 p = 3 1 Focus : 1 3 1, 7 Directrix : x = L : 1 3, 1 3 R : 1 3 1, Tim Howarth
28 Algebra II of 30. x 1 7 x + 1y 111 = 0 17 x 39 ( 7 x + 9 ) = 1y ( x 3 7 ) = 1y ( x 3 7 ) = 1( y 7 3) Vertex : 3, p = 1 p = Focus : 3 7, 3 3 Directrix : y = 11 3 L : 7, 3 3 R : 11 7, Note where each axis begins on the first graph. The second graph is if each axis was normal. y 1 x y + 7 = ( y y ) = 1 x ( y + 3) = 1 x 1 3 ( y + 3) = 1 x 7 Vertex : 7, 3 p = 1 p = 1 1 Focus : 7 1 1, 3 Directrix : x = 1 1 L : 7 1, 13 1 R : 7 1, Tim Howarth
29 Algebra II 9 of y + 1x 1 1 y = 0 1 ( y 1 1 y ) = 1x ( y 9 1 ) = 1x ( y 9 1 ) = 1( x + 7 ) Vertex : 7, 9 1 p = 1 p = 3 1 Focus : , 9 1 Directrix : x = L : , 1 1 R : , x + x y 3 = x + 1 ( x + 17 ) = 17 3 y ( x + 1 ) = 17 3 y ( x + 1 ) = ( y 1 3) Vertex : 1, 1 3 p = p = 1 Focus : 1, 1 Directrix : y = 9 L : 1 3 3, 1 R : 1 3, Tim Howarth
30 Algebra II 30 of x 1 1 x y + = x 1 1 ( x ) = 13 ( x ) = 13 y 0 11 ( x ) = 13 ( y ) Vertex :( 7 7 ), p = 13 p = Focus : 7 7, Directrix : y = 1 0 ( 0 ) 9 L :, R : 1 1, y y + 1 x + 7 y ( 7 y + 19 ) = 1 y = 0 ( y + 1 ) = 1 x ( y + 1 ) = 1 ( x 1 9 ) Vertex : 1 9, 1 p = 1 p = 9 3 Focus : 9, 1 Directrix : x = 113 L : 9, R : 9, x Tim Howarth
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