Download Exam Review Topics

Algebra II
Topics for 1st Semester Exam
1st 9-weeks:
Variables and expressions (base, power, exponent, evaluating for given values of
the variable)
Order of Operations (PEMDAS)
Categories of Numbers (Real, Rational, Irrational, Integers, Whole, Natural, Imaginary,
Complex (= Real + Imaginary))
Additive Identity ( a  0  a )
Multiplicative Identity ( a 1  a )
Multiplicative Property of Zero ( a 0  0 )
Additive Inverse Prop. ( a  ( a )  0 )
Properties:
Multiplicative Inverse Prop. (
a b
 1)
b a
Reflexive ( a  a )
Symmetric ( if a = b, then b = a)
Transitive (if a = b, and b = c, then a = c)
Substitution
Addition/Subtraction Properties of Equality
Multiplication/Division Properties of Equality
Trichotomy Prop. (exactly one of these is true:
either
ab
or
ab
ab
BIG 3:
Distributive a(b  c)  ab  ac
Commutative a  b  b  a or a  b  b  a
(Order)
Associative a  (b  c)  (a  b)  c or (a  b)  c  a  (b  c) (Grouping)
Solving Linear Equations
- using addition and subtraction
- using multiplication and division
Questions to ask yourself:
1. What is being done to the variable?
2. How can I “un-do” that?
3. Perform operation.
4. Check solution.
Solving Multi-step equations (work backward through the Order of Operations)
Solving equations with variable on both sides (get all variables on one side, all constants
on the other)
Literal Equations (solving for one variable in terms of another)
Absolute Value Equations
Recall that absolute value is always positive!
Set up 2 cases: if the number inside the abs.value symbol is
positive OR if it’s negative
You must ALWAYS CHECK your solution (your correct solving MAY
produce an invalid (extraneous) solution).
Solving Inequalities
Remember that when you multiply or divide by a negative number, you
must flip the sign!
Show solutions in set notation.
Open circles vs. closed circles- bold arrows, when?
Solving Absolute Value Inequalities
Compound inequalities
AND ( , < )
vs.
OR (  , > ) (graph goes in 2 directions)
Set up 2 cases: if the number inside the abs.value symbol is
positive or if it’s negative
Working Backwards from Graph to state Absolute Value Inequality
Coordinate Geometry:
Cartesian coordinate plane, terms and definitions, labeling axes,
origin, quadrants, abscissa, ordinate, domain, range, mapping
Relation vs. Function- (vertical line test, discrete vs. continuous function)
Linear Equations: (graph is a line) (degree of polynomial must be 1)
Both graphing and writing equations of lines
Standard Form (Ax + By = C)
Slope-intercept form ( y  mx  b )
Point-slope form
y  y1  m( x  x1 )
Graphing: using slope and y-intercept
Using x-intercept and y-intercept
Using a T-table
Review Best practices for graphing
Perpendicular lines: slopes are negative reciprocals
Parallel lines: have equal slopes
HOY VUX
Graphing Linear Inequalities & Systems of Linear Inequalities
Graph line (solid or dotted?), shade on the appropriate side by picking a test point
(the origin works really well); the solution to the system is where the graphs overlap
*** (Not covered for Jan 2013 Exam) Linear Programming, including word
problems
Graph system, shading only the overlapping region (region of feasibility)
Potential max & min at the vertices of the region
Define function first based on variables determined by what you’re asked to find
Evaluate function to find maximum or minimum
Honors: step functions (such as greatest integer function)
f ( x)   x 
Other Piecewise functions and their graphs
Graphing Absolute Value Equations
Recall that the graph looks like a “V”,
- how “wide” or “narrow” depends on the coefficient
- what x- value is it “centered” around (horizontal shift); what is the
vertical shift? Negative coefficient flips graph over the x-axis (upside
down)
Distance Formula
(x2  x1 )2  (y2  y1 )2
You should already know these 2 formulas
 x1  x2 y1  y2 
,


Midpoint Formula
2 
 2
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2nd Nine Weeks
Monomials and Polynomials
Monomials: Multiplying (add powers)
Dividing (subtract powers)
Power to a Power (multiply exponents)
Zero Exponent (anything to the zero power is 1)
Negative Exponents a  n 
1
(note: a negative exponent does NOT
an
change the sign of the number! It changes it’s location (top or bottom))
Scientific Notation (be able to go from standard notation to scientific and back;
Find products and quotients of number expressed in scientific notation)
Polynomials: type (monomial, binomial, trinomial)
Degree (the degree of each monomial term is the sum of its exponentsThe degree of the polynomial is the biggest degree of all of the monomial
pieces).
Descending powers of a variable
Adding and subtracting polynomials (combine like-terms; add/subtract the
coefficients) Subtract by adding its opposite.
Multiplying a Polynomial by a Monomial (distribute)
Multiplying a Polynomial by a Polynomial (distribute as many times as
necessary)
To multiply two binomials together, FOIL (distribute twice)
“Special Products” (square of a sum, square of a difference, difference of
squares) – still, FOIL!
Factoring
Prime, composite numbers
Prime factorization
Greatest common factor (GCF)
Factoring = “un-Distributing”
7 Rules for Factoring:
1) GCF
2) Difference of Two Squares
3) Trinomial with Leading Coeff. = 1
4) Sum/Difference of Two Cubes
5) Perfect Square Trinomials
6) Factoring by Grouping (4 terms)
7) Leading Coefficient > 1 (Guess and check)
Factoring “Flow Chart”
Solving Equations by Factoring (using zero product property)
Radicals
Radical expressions
Rational Exponents
Solving Radical Equations and Inequalities
Complex Numbers a  bi
Cycle of powers of i
Quadratics:
Solving Quadratic Equations by Factoring
Solving Quadratic Equations by Completing the Square
Quadratic Formula and the Discriminant
Graphing Quadratic “families”: parabolas:
2
vertical
y  a(x  h)  k
vertex at  h, k  , axis of symmetry x  h
Not responsible for horizontal parabolas right now
2
horizontal x  a(y  k )  h vertex at  h, k  , axis of symmetry y  k
More on Parabolas:
Form of equation
Axis of symmetry
Vertex
Focus
Directrix
Direction of opening
Length of latus rectum
vertical
y  a ( x  h) 2  k
xh
( h, k )
1 

 h, k 

4a 

1
yk
4a
Upward if a > 0
Downward if a < 0
1
units
a
horizontal
x  a( y  k ) 2  h
yk
( h, k )
1 

,k 
h
4a 

1
x  h
4a
Right if a > 0
Left if a < 0
1
units
a
Sum and product of the roots
Applications of Quadratics (word problems) – max if parabola opens down,
min if parabola opens up
Quadratic Inequalities: use a test point to determine which side of the parabola to shade,
check to see whether “line” of your curve should be solid ( ,  ) or dotted (< , >)
Solving Systems (show solutions as ordered pairs)
Graphing
Substitution
Elimination
Honors topic-Cramer’s Rule (set up coefficient matrix and substitute your constants in
place of the variable you’re solving for in numerator)
Terminology: Independent, Dependent, Inconsistent
Recall “special cases”: (CD = same line, infinite # solutions; Incon.= no solution =
parallel lines)
Solving Linear Systems and Non-Linear Systems, both graphically and algebraically
(specifically Linear-Quadratic and Quadratic-Quadratic)
New this year: for ALL graphs, be able to identify domain, range, for what
domain values the function is increasing, for what domain values the
function is decreasing
Describe the end behavior of a function (what is happening at the left end?
Right end?)
Additional Honors Topics:
More extensive work with rational exponents
Deriving Quadratic Formula
Simplifying radical expressions involving the use of absolute value symbols