Algebra II Honors

Parent Functions
1. Exponential Functions
  1. Compound Interest
    1. A(t)=P(1+r/n)^nt
  2. Half-life
    1. N(t)=initial amount(1/2)^t/d
  3. Double time
    1. N(t)=initial amount(2)^t/d
  5. b^x=a
2. Logarithmic Functions
  1. Inverse of Exponential Functions
    1. ln is the inverse of e
      1. lnx=a = e^a=x
  2. logb^a=x
  3. If the graph doesn't pass the horizontal line test there is no inverse equation
    1. Inverse is graphed reflected over line x=y
  4. Simplifying rules
    1. b^logb^x=logb^(b^x)
    2. logb^mn= logb^m x logb^n
    3. logb^m/n= logb^m-logb^n
    4. logb^(m^k)= k x logb^m
    5. logb^a= logb^a/logc^b
  5. Continually Compounded Interest A(t)=Pe^rt
3. Quadratic Functions
  1. Graphing
    1. Vertex Form: y=a(x-b)^2+k
    2. Standard form: ax^2+bx+c
    4. Vertex is the start of the function on the graph
    5. Projectile motion
      1. h(t)=1/2at^2+initial v+ initial h
  2. Factoring
    2. Shortcuts
      1. Perfect Square Trinomial a^2+2ab+b^2=(a+b)(a+b)
      2. Difference of Squares a^2-b^2=(a+b)(a-b)
    3. F.O.I.L.
  3. Quadratic Formula
    1. www.youtube.com/watch?v=O8ezDEk3qCg (go to this link for the song)
    2. Use to find roots/zeros/solutions/x-intercepts
    3. Negative under the radical creates 'i'
4. Polynomial Functions
  1. Multiplicity:The number of times a root is a solution
    1. If odd the graph passes through the x-axis
    2. If even the graph touches the x-axis
    3. Negative coefficients flip the graph and even number exponents make both sides go the same way while odd number exponents put them opposite ways
  2. Fundamental Theorem of Algebra tells how many roots there are
  3. Multiplying Polynomials
    1. Use box method
  4. Polynomial shortcuts
    1. Sum of Cubes
      1. a^3+b^3=(ab)(a^2-ab+b^2)
    2. Difference of Cubes
      1. a^3-b^3=(a-b)(a^2+ab+b^2)
  5. Steps to solve a polynomial 1.GCF 2. Shortcuts 3.Factor like a quadratic 4. Grouping
  6. Finding missing paranthesis
    1. 1.Find other roots on a calculator then use them to create other parenthesis 2.Divide by the original polynomial
5. Linear Functions
  1. Slope Intercept: y=mx+b
    1. Easier to find slope, y-intercept, and can be graphed
    2. Use to find line of best fit
  2. Standard Form: Ax+By=C
    1. Easier to find points, gives x and y intercepts, and nicer numbers
  3. Slope= rise/run
    1. Same between any two points
  4. Point Slope Form: y-y1=m(x=x1)
  5. Parallel lines have same slope
    1. Perpendicular lines have opposite reciprocal slope
      1. EX, 3/8x is perpendicular to -8/3x
6. Domain= all x values that are true Range= all y values that are represented
7. Function: one input has only one output
8. Transformations: Horizontal reflections= f(-x), Vertical reflections= -f(x), Horizontal translation=f(x+a), Vertical translation f(x)+b, Stretch or shrink c{f(x)}
Modeling with Equations
1. Order Of Operations
  1. 1. Paranthesis
    1. 2. Exponents
      1. 3. Multiply
        4. Divide
        5. Add
        6. Subtract
2. Define Variables
  1. EX. x= # of people, y=# of pets
    1. x+y=z
    2. Use equation to draw a picture of the scenario
3. Steps to model an equation
  1. 1.Identify Key Info/ What is the question
    1. 2. Define Variables
      1. 3. Write word equation
        4. Word equation--> Math equation
        5. Solve and Interpret Info
4. Graphing
5. Used in all units
Absolute Value
2. |x-a|<b
  1. a=midpoint b= distance endpoint to midpoint
  2. models range of error
3. separate problem into a positive and a negative then solve both
  1. check for extraneous solutions
Irrational and Complex Numbers
2. Conjugate: the pair to a binomial containing i or a radical with opposite signs and the same value. They rationalize the denominator because two radicals multiplied together cancel the radicals out, also making i^2=1
3. Solving Equations with Radicals
  1. One Radical
    1. 1. Isolate the Radical 2.Apply the inverse 3.Solve for x 4.Check for extraneous solutions
  2. Two Radicals
    1. 1. Put Radicals on either side 2.Apply the inverse 3.Combine like terms 4.Repeat if necessary 5.Check for extraneous solutions
4. Addition and subtraction with i
  1. Add or subtract terms without i
  2. Add or Subtract terms with i
  3. +
5. Multiplying with i
6. Dividing with i uses conjugate
7. Discriminant is the radical part of the quadratic formula
  1. A negative in the discriminant has an i
    1. Cannot be seen on graph
  2. A positive in the discriminant is a regular zero
    1. 2 solutions on graph
  3. A zero in the discriminant has one solution which is the origin
8. i^2=1
9. i is an imaginary number
Matrices
1. A scalar is distributed to each number in a matrix
2. A set of numbers in brackets
3. Addition and Subtraction of Matries
  1. Only between same dimensions. Add or subtract the numbers in the corresponding spaces in the matrices
4. Multiplication of Matrices
  1. Allowed only when the middle numbers of the dimensions are the same
    1. Resulting dimensions will be the outer two dimensions of the two matriceses being multiplied
  2. Multiply the first row by all the columns then he next by all the columns. The first row and first column numbers will be in the first row and first column etc.
  3. The determinate of a matrix is the top left through bottom right multiplied then subtracted by the other sets that were multiplied
  4. The inverse is one divided by the determinate multiplied by the matrix with the top left and bpttom right positions switched and the other two's signs switched
    1. Used to find x and y in a equation by multiplying the inverse by the two solutions to the equations
Systems
1. Substitution
  1. Solve one equation for y or x then use the y or x value to solve
2. Elimination
  1. Subtract two systems to cancel out a variable to solve for the other then use the new value to solve the other variable
3. Linear Programming
Proofs
1. Axioms: rules of math
2. Theorem: must be proven to be used
3. Truth Tables
  1. ^=conjunction (and) v=disjunction (or) -=negative -->= conditional <-->=biconditional
4. Counterexample: proves something untrue
5. Conditional: If A, then B
  1. A=hypothesis B=conclusion
  2. Converse If B, then A
  3. Biconditional: If and only if. Works both ways A<-->B
Inequalities
1. Compound Inequalities
  1. Conjuctions
    1. Uses the word "and"
      1. 4< 8x-12 and 8x-12<12
        2<x<6
  2. Disjunctions
    1. Uses the word "or"
      1. -2+8y<34 or -7-3y<14
        y<7 or y>7
  3. Number lines
2. Simple Ineaqualities in one variable
  1. EX. x+2<4 =x<2
3. Symbols
  1. _<. >_ are inclusive
    1. closed circle on a number line, dotted line on a graph
  2. <,> are exclusive
    1. open circle on a number line, regular line on a graph
Probability
1. Fundamental Counting Principle
2. Combination: order does not matter
  1. nCr=n!/r!(n-r)!
3. Permutation: order does matter
  1. nPr=n!/(n-r)!
    1. Repeated Elements P=n!/n1!n2!n3!......
4. Sample Size (n): All possiblities
5. P(event)=# of successes/ # of possibilities
6. Intersection (and) is when both events happen
7. Union (or) is when one or the other event happens
8. If A and B= P(A) x P(B) then they are independent. If not then they are dependent
9. 1-P(A)= the Complement
  1. Probability of the even not happening

Next up

Algebra II Honors

Description

Resource summary

Similar

	Created by 16jlangenfeld almost 11 years ago