Download Algebra 2

SOL Algebra 2
Properties
Commutative (order), Associative (grouping), Additive Identity (+ 0),
Additive Inverse, Multiplicative Identity (x 1), Distributive Property
Reflexive, Symmetric, & Transitive Properties of Equality
(Transitive – Inequality) If a > b and b > c, then a > c.
Addition Property, Subtraction Property, Multiplication Prop.
If a = b, then a + c = b + c.
If x + 1 > y, then x + 1 + z > y + z. (for inequality)
Simplifying Expressions
Imaginary numbers: i, i2 = -1, i3 = -i, i4 = 1
pattern
Factoring (top top, bottom bottom, cancel top & bottom)
Basic Exponent Laws (add or subtract exponents, distribute)
Factoring techniques:
CMF: Factor out what is common among terms.
DOTS: a2 – b2 = (a + b)(a - b)
SOTC: a3 + b3 = (a + b)( - + )
DOTC: a3 – b3 = (a - b)( + + )
Easy T (factors of last term that add up to middle term)
Perfect Square T: (x – 3)2 or (x + 5)2  not the same
as (x2 + 52)
Hard T (guess and check, or grouping)
Convert: Rational Exponent to Radical (root) Expressions
TI – 84: Dealing with imaginary numbers: Use (a + bi) MODE.
Use the exponent (^) key and USE ( ) for fractions.
Radical expressions (MATH menu: cube root, others)
The STORE key may be helpful to check your answers.
Parabola: y = a (x – h)2 + k
Technique: Find the vertex (h, k)
a > 0 (up)
a < 0 (down)
OR y – k = a (x – h)2
For horizontal parabolas: x = a (y – k)2 + h
OR x – h = a (y – k)2
a > 0 (facing right)
a < 0 (facing left)
SOL Algebra 2
Polynomials
x-intercepts  zeros  solutions (roots)
Graph
Function
Equation
If k is a zero, then f(k) = 0. When k is plugged into x, the answer is 0.
If 5 is a zero, then (x – 5) is a factor.
If (x + 3) is a factor, then –3 is a zero.
Think opposite!
If ½ is a zero, then (x – ½ ) is a factor. Using SLIDE, this becomes
the factor (2x – 1).
If -3/4 is a zero, then (x + ¾) is a factor. Using SLIDE, this becomes
the factor (4x + 3).
FOIL the factors to get the function: y = f(x) or equation: f(x) = 0.
# of turns + 1 = degree of function (linear, quadratic, cubic, quartic)
Double roots (graph merely touches the x-axis)
no real zeros  graph doesn’t touch or cross the x-axis
a > 0 (upward)
a < 0 (downward)
TI-84: Y=…, Graph, 2nd Trace [Calculate]
Use the STORE key to determine if a number is a zero or not.
For easy input, try 2nd Enter [Entry].
Don’t Forget: Parentheses, Parentheses, Parentheses
Functions
Function value, domain & range, operations including composition
TI-84: Y = …, GRAPH, TABLE, TABLESET (ASK)…
Pick a point (x,y) from the graph. Plug in the x & y and see if the
equation is true. Eliminate choices where equation is false. Pick another
easy point (x,y) if necessary. The x- and y- intercepts are good choices.
Transformations of Basic Graphs: effects of h & k
Horizontal shift: h (left or right)
Vertical shift: k (up or down)
Example: y = /x/ (basic graph with vertex at origin)
y = a /x – h/ + k (absolute value function)
Or y – k = a /x – h/  Think opposite…(h,k)  new vertex
SOL Algebra 2
Solving Equations
Use the STORE key. Check to see if there are 2 or more solutions.
Use (
Type in the left side and/or right side of the equation carefully.
) with fractions, rational expressions, quantities inside a square root.
If the discriminant b2 – 4ac for a quadratic equation is negative,
then the solutions are imaginary (involves i).
Rational Expressions: You can also use the LCD or crossmultiplication methods.
Radical Expressions: You can square both sides of EQ at the right
time (isolate radical first). Remember if the root is even, put + and -.
Quadratic Expressions: You can factor the left side or use the
quadratic formula. Get the standard form first: ax2 + bx + c = 0.
Absolute Value Equations/Inequalities
/2x + 1/ = 7
, 
, 
This branches out into two equations.
2x + 1 = 7
or
2x + 1 = -7.
Solve for x. The graph consists usually of two points.
endpoints have holes
full endpoints (dots)
/2x + 1/ > 7
(same with )
The graph consists of two separate parts.
Solution has the word ‘or’. Try solving: 2x + 1 > 7.
/2x + 1/ < 7
(same with )
The graph consists of one major part (sANDwich).
Solution looks like: a  x  b. Solve: -7 < 2x + 1 < 7.
TI-84 keys:
Y= … MATH NUM ABS for absolute value symbol
2nd MATH [TEST] for inequality symbols
GRAPH
SOL Algebra 2
Scatterplots/Linear Regression/Best Fit
Is there a correlation at all? Positive (rising line), Negative (falling
line), None (scattered)
Slope: m = rise / run
Y-intercept  b
Is it a line that best fits the data points? Then think of y = mx + b.
Also, draw a line that fits the data. Then read the coordinates.
Is it a curve (parabola) that fits the data? Then think of y = ax2+bx+c
Sometimes, you have to change the equation first. (Isolate y on the left.)
a > 0 (parabola facing up) a < 0 (down)
TI-84: Think of the right entries for L1 & L2. You may simplify
entries for L1. (example: number of years since 1980)
STAT EDIT …(input)… STAT CALC 4 [LinReg ax+b]
y = ax + b gives the equation from which you can predict y
Just plug number into x! L1  independent variable
L2  variable to predict or find
Systems of Equations/Inequalities
Look for intersection points of 2 graphs. Or solve the system.
(You usually solve for y first in terms of x.)
Input: Y1= … Y2 = …. GRAPH. Look at the coordinates of
intersection points. Or try 2nd TRACE [Calculate] Intersection
Linear Programming: Get the objective function. Plug in x & y
coordinates of vertices. Check what is asked for: maximum or minimum?
  solid line
Use y = mx + b form instead of standard
> or < dashed line
> upper portion shaded
< lower portion
Sequences & Variation
Sigma notation, arithmetic (d), geometric (r) patterns
Problem solving, arithmetic & geometric means, rule for an
Direct
vs.
Inverse
Joint: z = kxy
y = kx
y=k/x
(square, cube roots, etc.)
x/y=x/y
xy = xy
Word Problems (direct: divide & inverse: product = product)
REVIEW your Sequences Quiz and STATISTICS QUIZ!