Algebra 2 Trig AAlgebra 2 Trig A

Final Review 2007Final Review 2007

#1#1

HyperbolaCenter (0, 0)a = 8, b = 7, c =Vertices: (+8, 0)Foci: ( , 0)Slopes of asymptotes: +7/8

HyperbolaCenter (0, 0)a = 8, b = 7, c =Vertices: (+8, 0)Foci: ( , 0)Slopes of asymptotes: +7/8

x 2

64y 2

491

113

113

#2 y2 = 121 - x2#2 y2 = 121 - x2

Circle: x2 + y2 = 121Center: (0, 0)Radius = 11

Circle: x2 + y2 = 121Center: (0, 0)Radius = 11

#3 y = 2(x - 2)2 + 1#3 y = 2(x - 2)2 + 1

ParabolaCenter/Vertex: (2, 1)AOS: x = 2DOO: upFocus: (2, 9/8)Directrix: y = 7/8

ParabolaCenter/Vertex: (2, 1)AOS: x = 2DOO: upFocus: (2, 9/8)Directrix: y = 7/8

#4 6x2 + 16y2 = 96#4 6x2 + 16y2 = 96

Ellipse: Center: (0, 0)a = 4, b = , c =M vertices: (±4, 0)Foci: ( , 0)LMA = 8lma =

Ellipse: Center: (0, 0)a = 4, b = , c =M vertices: (±4, 0)Foci: ( , 0)LMA = 8lma =

x 2

16y 2

61

6

10

10

2 6

#5 x2 - 2x + y - 8 = 0#5 x2 - 2x + y - 8 = 0

Parabola: y = -(x - 1)2 + 9Center/Vertex: (1, 9)AOS: x = 1DOO: downFocus: (1, 8 3/4)Directrix: y = 9 1/4

Parabola: y = -(x - 1)2 + 9Center/Vertex: (1, 9)AOS: x = 1DOO: downFocus: (1, 8 3/4)Directrix: y = 9 1/4

#6 x2 = 2x + y2 - 4y + 7#6 x2 = 2x + y2 - 4y + 7

Hyperbola

Center: (1, 2)a = 2, b = 2, c = Vertices: (3, 2), (-1, 2)Foci: (1± , 2)Slopes of Asymptotes: ±1

Hyperbola

Center: (1, 2)a = 2, b = 2, c = Vertices: (3, 2), (-1, 2)Foci: (1± , 2)Slopes of Asymptotes: ±1

2 2

2 2

x 1 2

4y 2 24

1

#7 x2 +4y2 + 2x - 24y + 33 = 0#7 x2 +4y2 + 2x - 24y + 33 = 0

Ellipse

Center: (-1, 3)a = 2, b = 1, c =Vertices:(-3, 3),(1, 3)Foci: LMA = 4lma = 2

Ellipse

Center: (-1, 3)a = 2, b = 1, c =Vertices:(-3, 3),(1, 3)Foci: LMA = 4lma = 2

x 1 2

4y 3 21

1

3

1 3,3

#8 x2 + y2 = x + 2#8 x2 + y2 = x + 2

Circle

Center: (1/2, 0)Radius= 3/2

Circle

Center: (1/2, 0)Radius= 3/2

x 1

2

2

y 2 9

4

#9 Find f(x) + g(x)#9 Find f(x) + g(x)

f(x) = x2-x+3 g(x) = x+8

f(x)+g(x) = (x2-x+3) + (x+8)f(x)+g(x) = x2 + 11

f(x) = x2-x+3 g(x) = x+8

f(x)+g(x) = (x2-x+3) + (x+8)f(x)+g(x) = x2 + 11

#10 Find f(x) - h(x)#10 Find f(x) - h(x)

f(x) = x2-x+3 g(x) = x+8

f(x) - h(x) = (x2 - x + 3) - (3x2+1)f(x) - h(x) = x2 - x + 3 - 3x2 - 1f(x) - h(x) = -2x2 - x + 2

f(x) = x2-x+3 g(x) = x+8

f(x) - h(x) = (x2 - x + 3) - (3x2+1)f(x) - h(x) = x2 - x + 3 - 3x2 - 1f(x) - h(x) = -2x2 - x + 2

#11 Find f(g(x))#11 Find f(g(x))

f(x) = x2-x+3 g(x) = x+8 f(x) = x2 - x + 3f(g(x)) =(x+8)2 - (x+8) + 3f(g(x)) = x2 + 16x +64 - x - 8 + 3f(g(x)) = x2 +15x + 59

f(x) = x2-x+3 g(x) = x+8 f(x) = x2 - x + 3f(g(x)) =(x+8)2 - (x+8) + 3f(g(x)) = x2 + 16x +64 - x - 8 + 3f(g(x)) = x2 +15x + 59

#12 Find f(h(x))#12 Find f(h(x))

f(x) = x2-x+3 h(x) = 3x2+1

f(x) = x2 - x + 3f(h(x)) = (3x2+1)2 - (3x2+1) + 3f(h(x)) = 9x4+6x2+1-3x2-1+3f(h(x)) = 9x4+3x2+3

f(x) = x2-x+3 h(x) = 3x2+1

f(x) = x2 - x + 3f(h(x)) = (3x2+1)2 - (3x2+1) + 3f(h(x)) = 9x4+6x2+1-3x2-1+3f(h(x)) = 9x4+3x2+3

#13 Find g(f(x))#13 Find g(f(x))

g(x) = x+8 f(x) = x2-x+3

g(x) = x + 8g(f(x)) = (x2 - x + 3) + 8g(f(x)) = x2 - x + 11

g(x) = x+8 f(x) = x2-x+3

g(x) = x + 8g(f(x)) = (x2 - x + 3) + 8g(f(x)) = x2 - x + 11

#14 Find h(f(x))#14 Find h(f(x))

h(x) = 3x2+1 f(x) = x2-x+3

h(x) = 3x2 + 1h(f(x))= 3(x2 - x + 3)2 + 1h(f(x))= 3(x4-2x3+4x2 -3x+9)+1h(f(x))= 3x4-6x3+21x2-18x+27+1h(f(x))= 3x4-6x3+21x2-18x+28

h(x) = 3x2+1 f(x) = x2-x+3

h(x) = 3x2 + 1h(f(x))= 3(x2 - x + 3)2 + 1h(f(x))= 3(x4-2x3+4x2 -3x+9)+1h(f(x))= 3x4-6x3+21x2-18x+27+1h(f(x))= 3x4-6x3+21x2-18x+28

#15 Find h(g(x))#15 Find h(g(x))

h(x) = 3x2+1 g(x) = x+8

h(x) = 3x2 + 1h(g(x)) = 3(x + 8)2 + 1h(g(x)) = 3(x2 + 16x + 64)+1h(g(x)) = 3x2 + 48x + 192 + 1h(g(x)) = 3x2 + 48x + 193

h(x) = 3x2+1 g(x) = x+8

h(x) = 3x2 + 1h(g(x)) = 3(x + 8)2 + 1h(g(x)) = 3(x2 + 16x + 64)+1h(g(x)) = 3x2 + 48x + 192 + 1h(g(x)) = 3x2 + 48x + 193

#16 Find f(-3)#16 Find f(-3)

f(x) = x2 - x + 3

f(x) = x2 - x + 3f(-3) = (-3)2 - (-3) + 3f(-3) = 9 + 3 + 3f(-3) = 15

f(x) = x2 - x + 3

f(x) = x2 - x + 3f(-3) = (-3)2 - (-3) + 3f(-3) = 9 + 3 + 3f(-3) = 15

#17 Find h(f(4))#17 Find h(f(4))

h(x) = 3x2+1 f(x) = x2-x+3

f(4) = (4)2 - (4) + 3f(4) = 15h(x) = 3x2 + 1h(15) = 3(15)2 + 1h(f(4)) = 676

h(x) = 3x2+1 f(x) = x2-x+3

f(4) = (4)2 - (4) + 3f(4) = 15h(x) = 3x2 + 1h(15) = 3(15)2 + 1h(f(4)) = 676

#18 Find g(h(2))#18 Find g(h(2))

g(x) = x+8 h(x) = 3x2+1

h(2) = 3(2)2 + 1h(2) = 3(4) + 1h(2) = 13g(13) = 13 + 8g(h(2)) = 21

g(x) = x+8 h(x) = 3x2+1

h(2) = 3(2)2 + 1h(2) = 3(4) + 1h(2) = 13g(13) = 13 + 8g(h(2)) = 21

#19 Inverse of f(x) = 4x + 5#19 Inverse of f(x) = 4x + 5

y = 4x + 5x = 4y + 5x - 5 = 4yx/4 - 5/4 = y

y = 4x + 5x = 4y + 5x - 5 = 4yx/4 - 5/4 = y

f 1(x)1

4x 5

4

#20 Inverse of g(x) = 3x2 - 12#20 Inverse of g(x) = 3x2 - 12

y = 3x2 - 12x = 3y2 - 12x + 12 = 3y2

x/3 + 4 = y2

y = 3x2 - 12x = 3y2 - 12x + 12 = 3y2

x/3 + 4 = y2

g 1(x)1

3x 4

g 1(x)x 123

#21 f(x)=1/2x+2 g(x)=2x-4#21 f(x)=1/2x+2 g(x)=2x-4

f(g(x))=1/2(2x - 4) + 2f(g(x)) = x - 2 + 2f(g(x)) = x

f(g(x))=1/2(2x - 4) + 2f(g(x)) = x - 2 + 2f(g(x)) = x

#22 f(x) = 3x-9 g(x) = -3x+9#22 f(x) = 3x-9 g(x) = -3x+9

f(x) = 3x-9y = 3x - 9x = 3y - 9x + 9 = 3yx/3 + 3 = yNot equal to g(x)

f(x) = 3x-9y = 3x - 9x = 3y - 9x + 9 = 3yx/3 + 3 = yNot equal to g(x)

#23 {(1,3),(1,-1),(1,-3),(1,1)}#23 {(1,3),(1,-1),(1,-3),(1,1)}

{(3,1),(-1,1),(-3,1),(1,1)}Domain: 3, -1, -3, 1Unique x - coordinates

{(3,1),(-1,1),(-3,1),(1,1)}Domain: 3, -1, -3, 1Unique x - coordinates

#24 Simplify#24 Simplify

Simplify:Simplify:

x 2 4x 4x 2 x 6

3x 2 x 10x 2 9

x 2 2

x 3 x 2 x 3 x 3 x 2 3x 5

x 33x 5

#25 Simplify#25 Simplify

SimplifySimplifyb3

42a2

b2b2

a2

b

21

1b2

1

b3

2

#26 Simplify#26 Simplify

Simplify: Simplify:

7

2x 2y

3

x 2 2xy y 2

7 x y 2 x y x y

3 2 2 x y 2

7x 7y 62 x y 2

#27 Simplify#27 Simplify

Simplify: Simplify:

x 2 4x 2 2x 1x 2x 1

x 2 x 2 x 1 2

x 1x 2

x 2x 1

#28 Absolute value equation#28 Absolute value equation

Solve:Solve:

2 4x 2 19

2 4x 2 10

4x 2 54x 254x 3

x 3

4

4x 2 54x 7

x 7

4

#29 Absolute Value Inequality#29 Absolute Value Inequality

Solve:Solve:

3 x 1 12

x 1 4 4 x 14 3x 5

#30 Find f(-5)#30 Find f(-5)

If f(x) = 4x3 - x + 1f(-5) = 4(-5)3 - (-5) +1f(-5) = -500 + 5 + 1f(-5) = -494

If f(x) = 4x3 - x + 1f(-5) = 4(-5)3 - (-5) +1f(-5) = -500 + 5 + 1f(-5) = -494

#31 Do the math#31 Do the math

(8x3 + 2x2 + 3x)÷(2x + 3)(8x3 + 2x2 + 3x)÷(2x + 3)

2x 3 8x 3 2x 2 3x

4x 2 5x 9 27

2x 3

#32 Simplify#32 Simplify

Simplify:Simplify:

36x 6y10

6xy 6

6x 5y16

#33 Factor: 27a3 + 125b3#33 Factor: 27a3 + 125b3

Factor: 27a3 + 125b3

(3a + 5b)(9a2 - 15ab + 25b2)

Factor: 27a3 + 125b3

(3a + 5b)(9a2 - 15ab + 25b2)

#34 Factor: 9x2 - 12x + 4#34 Factor: 9x2 - 12x + 4

Factor: 9x2 - 12x + 4(3x -2)2

Factor: 9x2 - 12x + 4(3x -2)2

#35 Factor: 7y - 12x + 4xy - 21#35 Factor: 7y - 12x + 4xy - 21

Factor: 7y - 12x + 4xy - 217y - 21 + 4xy - 12x7(y - 3) + 4x(y - 3)(y - 3)(7 + 4x)

Factor: 7y - 12x + 4xy - 217y - 21 + 4xy - 12x7(y - 3) + 4x(y - 3)(y - 3)(7 + 4x)

#36 Factor: 15a3b - 5a2b2 - 10ab3#36 Factor: 15a3b - 5a2b2 - 10ab3

Factor: 15a3b - 5a2b2 - 10ab3

5ab(3a2 - ab - 2b2)5ab(3a2 - 3ab +2ab - 2b2)5ab[3a(a - b) + 2b(a - b)]5ab(a - b)(3a + 2b)

Factor: 15a3b - 5a2b2 - 10ab3

5ab(3a2 - ab - 2b2)5ab(3a2 - 3ab +2ab - 2b2)5ab[3a(a - b) + 2b(a - b)]5ab(a - b)(3a + 2b)

#37 Simplify: #37 Simplify:

Simplify:Simplify:

64a5b33

64a5b33

4 3a3a2b33

4ab a23

#38 Simplify:#38 Simplify:

Simplify:Simplify:

9a 1

5 a4

5

9a 1

5 a4

5

91

5 a5

5

91

5 a

#39 Simplify:#39 Simplify:

Simplify:Simplify:

5 27a3 64a73

3 73

33 3 23

32 2

32 2

5 27 64

5 3 4

5 3 4

60

a a

a a a

a a

a a

#40 Solve:#40 Solve:

Solve:Solve:

a1

5 20

a1

5 20

a1

5 2

a25

a32

#41 Solve: x2 + 441 = 0#41 Solve: x2 + 441 = 0

Solve: x2 + 441 =0x2 = -441x = x = ±21i

Solve: x2 + 441 =0x2 = -441x = x = ±21i

441

#42 Simplify: (9 - 3i) - (3 + 5i)#42 Simplify: (9 - 3i) - (3 + 5i)

(9 - 3i) - (3 + 5i)9 - 3 - 3i - 5i6 - 8i

(9 - 3i) - (3 + 5i)9 - 3 - 3i - 5i6 - 8i

#43 Simplify: (5 + 4i)(3 - 7i)#43 Simplify: (5 + 4i)(3 - 7i)

Simplify: (5 + 4i)(3 - 7i)(5 + 4i)(3 - 7i)15 - 35i + 12i - 28i2

15 - 23i - 28(-1)15 - 23i + 2843 - 23i

Simplify: (5 + 4i)(3 - 7i)(5 + 4i)(3 - 7i)15 - 35i + 12i - 28i2

15 - 23i - 28(-1)15 - 23i + 2843 - 23i

#44 Simplify: #44 Simplify:

Simplify:Simplify:

5

7 i

5

7 i5

7 i7 i7 i

5 7 i 49 i2

5 7 i 49 1

5 7 i 50

7 i10

#45 Simplify: (7 - 3i)(7 + 3i)#45 Simplify: (7 - 3i)(7 + 3i)

Simplify: (7 - 3i)(7 + 3i)49 + 21i - 21i - 9i2

49 - 9(-1)49 + 958

Simplify: (7 - 3i)(7 + 3i)49 + 21i - 21i - 9i2

49 - 9(-1)49 + 958

#46 Simplify: i10i21i30#46 Simplify: i10i21i30

Simplify: i10i21i30

i10+21+30 = i61 = i4(15)+1 = i1 = i

Simplify: i10i21i30

i10+21+30 = i61 = i4(15)+1 = i1 = i

#47 Simplify#47 Simplify

Simplify:Simplify:

4 i 74 i 74 i 74 i 7

4 i 74 i 7

16 8i 7 7i2

16 7i29 8i 723

#48 Solve: x2 + 5x + 13 = 0#48 Solve: x2 + 5x + 13 = 0

x2 + 5x + 13 = 0x2 + 5x + 13 = 0

x 5 52 4 1 13

2 1

x 5 25 52

2

5 272

53i 3

2

#49 Solve: 6x2 + 7x = 3#49 Solve: 6x2 + 7x = 3

Solve:Solve:

6x 2 7x 30

6x 2 9x 2x 30

3x 2x 3 1 2x 3 02x 3 3x 1 0

x 32,1

3

#50 Solve: 2x2 + 3x - 13 = 0#50 Solve: 2x2 + 3x - 13 = 0

Solve:Solve:

2x 2 3x 130

x 3 32 4 2 13

2 2 3 9 104

4

x 3 113

4

#51 Word Problem#51 Word Problem

h(t) = -16t2 + 10t + 50h(1) = -16(1)2 + 10(1) + 50h(1) = 44 feet0 = -16t2 + 10t + 50

h(t) = -16t2 + 10t + 50h(1) = -16(1)2 + 10(1) + 50h(1) = 44 feet0 = -16t2 + 10t + 50

t 10 102 4 16 50

2 16 1010 33

32t 2.107seconds

#52 Simplify, combine like terms#52 Simplify, combine like terms

(4b4 + 6b2 - 3b + 5) - (2b3 + 3b - 2)4b4 - 2b3 + 6b2 - 6b + 7

(4b4 + 6b2 - 3b + 5) - (2b3 + 3b - 2)4b4 - 2b3 + 6b2 - 6b + 7

#53 Simplify, remove parentheses#53 Simplify, remove parentheses

(y + 2)(y2 - 4y + 1)y3 - 4y2 + y + 2y2 - 8y + 2y3 - 2y2 - 7y + 2

(y + 2)(y2 - 4y + 1)y3 - 4y2 + y + 2y2 - 8y + 2y3 - 2y2 - 7y + 2

#54 Do the arithmetic#54 Do the arithmetic

(2x3 - 3x2 + 4x - 5) ÷ (x - 2)Synthetic Division2 2 -3 4 -5 4 2 12 2 1 6 7

2x2 + x + 6x + 7/(x-2)

(2x3 - 3x2 + 4x - 5) ÷ (x - 2)Synthetic Division2 2 -3 4 -5 4 2 12 2 1 6 7

2x2 + x + 6x + 7/(x-2)

#55 Factor#55 Factor

64x2y2 - 25z2

(8xy - 5z)(8xy + 5z)

64x2y2 - 25z2

(8xy - 5z)(8xy + 5z)

#56 Find the zeros#56 Find the zeros

y = 3x2 + 5x + 20 = 3x2 + 5x + 20 = (3x2 + 3x) + (2x + 2)0 = 3x(x + 1) + 2(x + 1)0 = (x + 1)(3x + 2)x = -1, -2/3

y = 3x2 + 5x + 20 = 3x2 + 5x + 20 = (3x2 + 3x) + (2x + 2)0 = 3x(x + 1) + 2(x + 1)0 = (x + 1)(3x + 2)x = -1, -2/3

#57 Find the max or min of #56#57 Find the max or min of #56

y = 3x2 + 5x + 2DOO: up, therefore a minimumx = -b/2a x = -5/2(3) = -5/6y = 3(-5/6)2 + 5(-5/6) + 2y = 25/12 - 25/6 + 2y = -1/12Vertex is (-5/6, -1/12)

y = 3x2 + 5x + 2DOO: up, therefore a minimumx = -b/2a x = -5/2(3) = -5/6y = 3(-5/6)2 + 5(-5/6) + 2y = 25/12 - 25/6 + 2y = -1/12Vertex is (-5/6, -1/12)

#58 Solve Systems of Equations#58 Solve Systems of Equations

x + y = 2x – 3y =6

-x-y=-2 x+(-1)=2x-3y=6 x=3 -4y=4y=-1

x + y = 2x – 3y =6

-x-y=-2 x+(-1)=2x-3y=6 x=3 -4y=4y=-1

#59#59

(3x-5y=6)*7 21x-35y=42(2x+7y=12)*5 10x+35y=60

(3x-5y=6)*7 21x-35y=42(2x+7y=12)*5 10x+35y=60

#60 Solve Systems of Equations#60 Solve Systems of Equations

7x+3y=-12x-y=9 y=2x-9

7x+3y=-12x-y=9 y=2x-9

#61 Solve a system#61 Solve a system

3x+2y=-24 6x-5y=30-6x-4y=48 6x-5y=30 -9y=78 y=-78/9

3x+2y=-24 6x-5y=30-6x-4y=48 6x-5y=30 -9y=78 y=-78/9

432

yx

165

yx

#62#62

Is (1, 2) a solution to the following system of inequalities?

x>1, y<3, y>2x-11>1, 2<3, 2>2(1)-1Yes. Yes. No.The answer is NO.

Is (1, 2) a solution to the following system of inequalities?

x>1, y<3, y>2x-11>1, 2<3, 2>2(1)-1Yes. Yes. No.The answer is NO.

END of ReviewEND of Review