Download Algebra 2 L to J 2008-2009

Algebra 2 L to J Essential Learning 2008-2009
1 Absolute value
Alg II a   a, a  0

2 Arithmetic sequence
Alg II A pattern of numbers where the common difference of consecutive terms is the same
3 Arithmetic series
Alg II The sum of the elements of an arithmetic sequence
4 Asymptote
Alg II The line that a curve approaches
5 Axis of Symmetry
Alg II The line that splits a graph into symmetrical parts
6 Complex conjugate
Alg II
7 Complex Fraction
Alg II A fraction that contains a fraction in the numerator, denominator or both
8 Complex Number
Alg II Any number of the form
9 Composition
Alg II
  a, a  0
(a  bi ) and (a  bi )
a  bi
where a,b are real numbers
f g ( x)  f [ g ( x)], and , g f ( x)  g[ f ( x)]
10 Consistent Dependent System Alg II A system of equations with an infinite number of solutions
11 Consistent Independent System Alg II A system of equations with exactly one solution
12 Constraints
Alg II Boundaries used in Linear Programming
13 Direct Variation
Alg II
14 Discriminant
Alg II
15 Feasible region
Alg II The area of intersection of a system of inequalities
16 Geometric Sequence
Alg II A sequence of numbers where each term is found by multiplying by a constant
17 Geometric Series
Alg II The sum of n terms of a geometric sequence
18 Imaginary Number
Alg II
19 Inconsistent System
Alg II A system of equations with no solution
y  kx
, x and y increase or decrease together
b2  4ac
, used to determine the number and type of solutions to a quadratic
equation
i
1
20 Inverse functions
Alg II A reflection over the line y = x
21 Inverse Variation
Alg II
22 Iteration
Alg II A function in which the output becomes the next input
23 Linear Regression
Alg II A line of best fit
24 Matrix
Alg II An organization of numbers or variables in columns and rows
25 Objective Function
Alg II The function being maximized or minimized in Linear Programming
26 Parabola
Alg II
27 Radical Expression
Alg II An expression with variables inside/under the radical
28 Rational Expression
Alg II A polynomial divided by a polynomial
29 Recursive Formula
Alg II In a sequence, shows how to find the nth term from the terms before or after it
30 Vertex
Alg II Maximum or minimum of a parabola
31 Zero product property
Alg II If ab=0, then a=0, b=0, or both a and b = 0
32 Zeros/Roots
Alg II The x - intercepts of a function
y
k
x as x increases y decreases and vice versa
y  a( x  h)2  k or x  a( y  k )2  h
L to J Concepts
Algebra II
Essential Learning
Example Problem with Solution
1. Point slope form of a line
y  y1  m( x  x1 )
2. Write the equation of a line given
two points.
( 2, 4 ) and ( 6, - 8)
y – 4 = -3( x – 2 )
3. Write the equation of a line given
one point and slope.
m = 8 point ( 3 , -5 )
(y + 5) = 8(x– 3 )
4. Direct Variation
y varies directly as x, when x = 5, y = 25
find y when x = 7 y  5 x
m = -3
y  35
5. Inverse Variation
x varies inversely as y, when y = 12, x = 10
find x when y = 4
6. Linear Regression
Write equation of line of best fit through (1,1),
(2,1),(3,3), and (4,4)
yx
y
5
7. Graphing Linear Equations
3
y   x5
4
4
3
2
1
x
-5 -4 -3 -2 -1
-1
1
2
3
4
5
-2
-3
-4
-5
8. Graphing Linear Inequalities
9. Graphing Greatest Integer Functions
y  2x 
3
2
y   x 1  2
y
10. Graphing Absolute Value Functions
5
y  x 3
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
x
1 2 3 4 5 6 7
-2
-3
-4
-5
11. Solve Compound Inequalities
8  2x  4  10
12. Solve Absolute Value Equations
2 x  3  18
13. Solve for Specified Variable
s1v 
14. Solve systems of equations by substitution.
2hf
5
s2
1
x3
2
y  2x 1
y
6  x  3
x
15
or
2

21
2
solve for f
f 
s1s2 v  5s2
2h
8 13
( , )
3 3
y
15. Solve systems of equations by graphing.
And identify the type of solution
5
y  x3
4
3
3 x  3 y  9
y  x3
2
1
x
-5 -4 -3 -2 -1
-1
1
2
3
4
5
Consistent
Dependent
-2
-3
-4
-5
y
x  2y 1
5
2x  y  5
3
4
2
1
x
-5 -4 -3 -2 -1
-1
1
2
3
4
(3,-1)
Consistent
Independent
5
-2
-3
-4
2 y  6x  6
-5
y
y  3x  1
5
4
3
2
1
-5 -4 -3 -2 -1
-1
x
1
2
3
4
5
No solution
Inconsistent
-2
-3
-4
-5
16. Solve systems of equations by elimination.
17. Graphing Systems of Inequalities
6 x  5 y  27
(2, 3)
3 x  10 y  24
x 1
y3
x y 5
18. Simplifying with Rule of Exponents
[(6 x 2 y 3 )]2
[(3x 1 y 2 )2 ]1
19. Multiply Polynomials
( x  4)( x  2)( x  1)
20. Divide Polynomials
( x 2  5x  6)  ( x  3)
21. Factor Polynomials
6m 2  7 m  3
y10
4 x6
x3  7 x 2  14 x  8
x2
(3m  1)(2m  3)
5(3v  5)(v  2)
15v 2  55v  50
x3  8
( x  2)( x 2  2 x  4)
49 x6  625 y 4
22. Add/Subtract Polynomials
(7 x3  25 y 2 )(7 x3  25 y 2 )
6 x3  x 2  3x  2
(3x 2  4 x  5)  (6 x3  2 x 2  x  3)
23. A third degree equation
x3  81
x  33 3
24. A fourth degree equation
x 4  625
x  5
25. Simplification of complex number
25  5i
20  2i 5
3 72  18i 2
26. Addition of complex numbers
(2 + 3i ) + ( 5 + 2i ) = 7 + 5i
(3 – 5i ) + ( 6 + i ) = 9 – 4i
27. Subtraction of complex numbers
(2 + 3i ) - ( 5 + 2i ) = -3 + i
(3 – 5i ) - ( 6 + i ) = -3 – 6i
28. Multiplication of complex numbers
( 6 + 2i )( 3 – 5i ) = 28 - 4i
( 3 + 2i ) 2 = 5 + 12i
29. Division of complex numbers
6  2i (6  2i)(3  i) 8  6i


3i
(3  i)(3  i)
5
30. Cyclic Powers of i
31. Solve by completing square
i 7 2 i
3x  6 x  14
32. Solve by using square roots
x2  4  0
33. Solving by quadratic formula
x2  8x  5  0
x
34. Solve using Quadratic Techniques
x4  5x2  6  0
x  i 2 or x  i 3
35. Use the Discriminant to Find the Type
of Roots and Number
x
x  2i or  2i
5 x 2  7 x  12
36. Graph y = 2(x – l )
-3
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
x
1
2
3
8  44
 4  11
2
discriminant = -191
2 imaginary roots
y
2
1  51
3
4
5
37. Determine if a relation is a function
38. Find f
g (3) if f ( x)  x 2 and g ( x)  x  1
39. Identify the domain and range of a function
{(0,2), (3,4), (0,5), (-12,-1)} --no
y = 3x – 1
--yes
vertical line test for functions
f g (3)  4
y  x  2 1
Domain {x : x  all reals}
Range { y : y  1}
y
5
4
3
40. Determine if a function has an inverse, if so
find it and graph it
y = 3x+2
1
2
f 1 ( x)  x 
3
3
y=3x+2
2
y = 1/3x - 2/3
1
-5 -4 -3 -2 -1
-1
x
1
2
3
4
5
-2
-3
-4
-5
41. Recursion and Iteration
42. Addition of radicals
Find the first three iterates
x1  16, x2  142, x3  1276
f ( x)  9 x  2, x0  2
34 3 5 3
6 5 2 5  4 5
8  18  2 2  3 2  5 2
43. Subtraction of radicals
3  4 3  3 3
6 5 2 5  4 5
44. Multiplication of radicals
8  18  2 2  3 2  1 2
3 * 5  15
3 2 *5 7  15 14
45. Division of radicals
6
 3
2
7
21

3
3
46. Radical equations
2x  3  6
2x  3  6
47. Simplify Rational Expressions
2x  x  3  1
x
3

x 3 x  2
3x  6 14 x  14
7 x  7 5 x  10
x2  2x  8 x  2

x 2  4 x  3 3x  3
x = 33/2
x = 9/2
x = 4 or 1
x2  5x  9
( x  3)( x  2)
3( x  4)
x3
6
5
48. Rational Equations
2x 2
 5
x 5 5
49. Arithmetic Sequences
an  a1  (n  1)d
50. Arithmetic Series
51. Geometric Sequence
52. Geometric Series
sn 
x
1
a1  5, d  , n  12, an  ?
3
26
a12 
3
7  14  21  28  ...  98
n
(a1  an )
2
S n  735
an  a1r n 1 or an  an 1r
a1 (1  r n )
Sn 
1 r
15
13
1
, n  9, r  4, an  ?
64
a9  1024
a1 
a1  5, r  2, n  14, Sn  ?
Sn  81,915