SOL STUDY GUIDE – ALGEBRA 2
PROPERTIES – SOL A2.1
Commutative ex.  a  b   c  c   a  b 
GRAPHS AND THEIR EQUATIONS –SOL A2.6
a bb a
y
x
Associative
ex.  a  b   c  a  b  c 
 a b c  a b c 
Distributive ex. a  b  c   ab  ac
Multiplicative Property of Zero ex. a 0  0
Multiplicative Identity ex. a 1  a
Additive Identity ex. a  0  a
Additive Inverse ex. a  (a)  0
a b
1
b a
Transitive ex. If a  b and b  c , then a  c
If a  b and b  c , then a  c
Reflexive ex. a  a
Symmetric ex. If a  b then b  a
Substitution ex. If a  b , then b can replace a
Equality ex. If a  b then a  c  b  c
If a  b then a  c  b  c
If a  b then a c  b c
If a  b then a  c  b  c
Linear: y  x
y
y
y
x
x
Quadratic: y  x
y  x
2
2
y
y
x
x
x
Polynomial: y  x3
y  x4
y  x5
y
y
x
x
Absolute value: y | x |
y  | x|
y
x
Greatest Integer (Step Function): y   x
Imaginary Numbers – 3i, 4
Real Numbers – Includes all number sets below
Irrational Numbers – 0.2346910…, 
Integers – …-3, -2, -1, 0, 1, 2, 3, …
Whole Numbers – 0, 1, 2, 3, …
Natural Numbers – 1, 2, 3, …
x y
y
Multiplicative Inverse ex.
3
Rational Numbers – 0.25, 0.141414…,
4
x
2
y
y
x
x
Square Root: y  x
y x
y
y
x
Exponential: y  b , b  1
x
y
x
Logarithmic: y  logb x
x
y  b ,b 1
x
SOL STUDY GUIDE – ALGEBRA 2
SOLVING
Absolute Value – SOL A2.4 There will be 2 possible solutions!
Isolate the absolute value then split into two equations, one
equation is negated.
Ex. 2  x  4  8
Ex. 3 x  2  12
x2 4
x4 6
x46
x  4  6
x  10
x24
x  2  4
x2
x  6
x  2
SYSTEMS OF EQUATIONS AND INEQUALITIES – SOL A2.12, A2.13
The answer to a system of equations is the ordered pair(s) where
the graphs intersect. Use substitution or elimination to solve
algebraically.
The answer to a system of inequalities is where the shaded regions
intersect.
Quadratics – SOL A2.6 Roots = Zeros = Solutions = X-Intercepts
Factoring: Always look for a GCF first!
Completing the Square: Don’t forget the  .
b  b2  4ac
(given on SOL formula sheet)
2a
Discriminant: b 2  4ac Helps determine the roots.
b 2  4ac > 0 gives 2 real solutions
b 2  4ac < 0 gives 2 complex (imaginary) solutions
b 2  4ac = 0 gives 1 real solution
Quadratic Formula:
DOMAIN AND RANGE – SOL A2.9
Domain: x values
Range: y values
Ex. {(2,3), (4,3), (1,2)} Domain={2,4,1} Range={3,2}
What is the domain and range of function graphed?
y
Square Root – SOL A2.3 There will be 2 solutions because of the  .
Ex.
Ex.
x 2  36
x 2  48
x  36
7
6
5
(–1, 3)
x  48
2
2
x  6
4
3
2
x  4 3
1
–7 –6 –5 –4 –3 –2 –1
–1
Squaring – SOL A2.7 Check your answer for extraneous solutions!
Ex.
Ex.
x2 5
42 x 8

x2

2
  5
x  2  25
x  23
2
2 x 4
 x
  2
x4
2
3
4
–3
–4
(–6, –5)
–5
–6
–7
x 2
2
1
–2
2
Domain: {x | -6 < x < -1}
Range: {y | -7 < x < 4}
5
6
7
x
SOL STUDY GUIDE – ALGEBRA 2
EXPONENT RULES – SOL A2.10
a 2  a 2  2a 2 can add only like terms, don't change exponent
CONICS – SOL A2.18
Formulas can also be found on the Casio!!
y
y
a (a )  a add exponents when multiplying
2
2
4
(a 2 ) 4  a8 multiply exponents when power of a power
a8
 a 3 subtract exponents when dividing
a5
a 0  1 anything to the zero power = 1
1
a 1  1
a
x
Parabola: y  ( x  h)  k
2
y
x
Circle: ( x  h)  ( y  k )  r
2
2
2
1
 a1 flip negative exponents to make positive
a 1
RADICALS AND RATIONAL EXPRESSIONS – SOL A2.3, A2.11
x
or x  ( y  k )  h
2
y
x
( x  h) 2 ( y  k ) 2
Ellipse:

1
a2
b2
y
3
Ex.
4
x 3  x 4 Exponent = Numerator and Index = Denominator
3
54 x 6 y 8  3x 2 y 2 3 2 y 2
5 2 3 5  4 2  9 2 3 5


2
2
Hyperbola: ( x 2h)  ( y 2k )  1
a
x
b
y
Only combine like terms.
2 3 5 2  3 5  10 6  6 15
COMPLEX NUMBERS (IMAGINARY) – SOL A2.3, A2.17
i 2  1 i 3  i
i4  1
MEMORIZE!!!! i1  i
FACTORING – SOL A2.5
Always look for a GCF first!!
Difference of squares: x 2  y 2  ( x  y)( x  y )
Difference of cubes: x3  y3  ( x  y)( x 2  xy  y 2 )
Sum of cubes: x3  y3  ( x  y)( x 2  xy  y 2 )
Trinomial: x 2  5x  6  ( x  3)( x  2) 2 x 2  3x  2  (2 x  1)( x  2)
( y  k ) 2 ( x  h) 2

1
a2
b2
x
SEQUENCE AND SERIES – SOL A2.16
Arithmetic: add the same difference to make the next term
Geometric: multiply the same factor to make the next term
 last

Series:   expression  ; Plug in the numbers from the first to the
 first

last into the expression, then add their values.
x4
Ex.
2
x 1
 (211 )  (221 )  (231 )  (241 )
x 1
 (20 )  (21 )  (22 )  (23 )
 1  2  4  8  15
SOL STUDY GUIDE – ALGEBRA 2
INVERSE – SOL A2.9
Switch x and y and re-solve for y!
Ex. y  5 x  8
x  5y  8
Inverse: x  8  5 y
x 8
y
5
SCATTER PLOTS AND LINEAR REGRESSION – SOL A2.19
Line of Best Fit – line that best represents the scatter plot
Equation of the Line of Best Fit
Calculator: Stat>enter data into List I and List2
>F1 (Grph)>F1 (GPH1)>F1 (x)
Use values given for a and b to write the equation in
y = ax + b form. Use the equation to make predictions.
COMPOSITION OF FUNCTIONS – SOL A2.9
Take the innermost function and substitute it in the function listed
to the immediate left.
Ex. Given q  x   4 x  7 and p  x   2 x  8 , find  p q  x  .
 p q  x   p  q  x  
 p  4x  7
 2  4x  7  8
 8x  6
VARIATION – SOL A2.20
k is the constant and is in all variation problems
k
Direct: y  kx
Inverse: y 
Joint: y  kxz
x
Pay attention to the order of the phrase “varies ________ as”.
Ex. If n varies directly as the cube of g. Find the constant of
variation if n = 32 when g = 2. Find g when n = 108.
n
32 108
32
Formula: k  3
 3
k 3
g
(2)3
g
(2)
k 4
32 g 3  864
g 3  27
g 3  27
g 3
RATIONAL EXPRESSIONS – SOL A2.2, A2,7
Follow the same fraction rules and factor polynomials, as needed, to
simplify.
x 2  3x  18  x  3 x  6 
Ex.

 x6
x 3
x 3
MATRICES – SOL A2.11
 2 6  1 5  1 1
Ex. 


 Need like dimensions.
 6 10   7 8 1 2 
 2 6  6 18 
3 

 Distribute scalar to all elements.
 6 10  18 30
 2 6
1 2 
   10 14 Columns of 1 must match rows of 2.
 6 10 
1 0 
0 1  Identity


3
 5

 

 2 6 
8
8 Inverse

 6 10    3
1

 

 8
8 