Algebra 2 Study Notes

Algebra 2 Study Notes

Key Concepts

• Functions

  • Definition: A relation where each input has a unique output.
  • Types: Linear, quadratic, polynomial, rational, exponential, logarithmic.

• Complex Numbers

  • Form: a + bi (where a and b are real numbers, i is the imaginary unit).
  • Operations: Addition, subtraction, multiplication, division.
  • Polar form: r(cos θ + i sin θ).

• Quadratic Functions

  • Standard form: ( ax^2 + bx + c ).
  • Vertex form: ( a(x-h)^2 + k ) (vertex at (h, k)).
  • Factoring: Can be factored into the form ( (px + q)(rx + s) ).
  • Quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

• Polynomials

  • Degree: Highest power of the variable.
  • Operations: Addition, subtraction, multiplication, division (synthetic division).
  • The Remainder Theorem: The remainder of f(x) divided by (x - c) is f(c).
  • The Factor Theorem: x - c is a factor of f(x) if f(c) = 0.

• Exponential and Logarithmic Functions

  • Exponential form: ( f(x) = a \cdot b^x ).
  • Logarithmic form: ( y = \log_b(x) ) implies ( b^y = x ).
  • Properties:
    • ( \log_b(xy) = \log_b(x) + \log_b(y) )
    • ( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) )
    • ( \log_b(x^n) = n \cdot \log_b(x) ).

• Systems of Equations and Inequalities

  • Solving methods: Graphing, substitution, elimination.
  • Types: Linear systems, nonlinear systems.
  • Inequalities: Solutions can be expressed in interval notation.

• Sequences and Series

  • Arithmetic sequence: ( a_n = a_1 + (n - 1)d ) (d = common difference).
  • Geometric sequence: ( a_n = a_1 \cdot r^{(n-1)} ) (r = common ratio).
  • Summation formulas:
    • Arithmetic series: ( S_n = \frac{n}{2}(a_1 + a_n) )
    • Geometric series: ( S_n = a_1 \frac{1 - r^n}{1 - r} ) (r ≠ 1).

• Conic Sections

  • Types: Circles, ellipses, parabolas, hyperbolas.
  • Standard equations:
    • Circle: ( (x-h)^2 + (y-k)^2 = r^2 )
    • Ellipse: ( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 )
    • Parabola: ( y = a(x-h)^2 + k )
    • Hyperbola: ( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 ).

• Probability and Statistics

  • Basic concepts: Mean, median, mode, range.
  • Probability rules: Conditional probability, independent events, permutations, combinations.

• Matrices

  • Definition: Rectangular array of numbers.
  • Operations: Addition, subtraction, multiplication, finding the determinant.
  • Applications: Solving systems of equations, transformations.

Important Skills

• Factoring polynomials and solving polynomial equations.
• Working with functions: transformations, compositions, and inverses.
• Solving equations involving exponents and logarithms.
• Analyzing and interpreting data sets.

Key Concepts

• Functions

  • A function is a relation where each input corresponds to one unique output.
  • Types of functions include linear, quadratic, polynomial, rational, exponential, and logarithmic.

• Complex Numbers

  • Expressed as ( a + bi ), where ( a ) and ( b ) are real numbers, and ( i ) is the imaginary unit.
  • Operations can be performed: addition, subtraction, multiplication, and division.
  • Polar form is represented as ( r(\cos \theta + i \sin \theta) ).

• Quadratic Functions

  • Standard form is ( ax^2 + bx + c ).
  • Vertex form is ( a(x-h)^2 + k ), with the vertex at the point (h, k).
  • Quadratic expressions can be factored into ( (px + q)(rx + s) ).
  • The quadratic formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

• Polynomials

  • The degree refers to the highest exponent of the variable in the polynomial.
  • Operations include addition, subtraction, multiplication, and division, with synthetic division as a specific technique.
  • The Remainder Theorem states that the remainder of ( f(x) ) divided by ( (x - c) ) equals ( f(c) ).
  • The Factor Theorem indicates that ( x - c ) is a factor of ( f(x) ) if ( f(c) = 0 ).

• Exponential and Logarithmic Functions

  • Exponential form is given by ( f(x) = a \cdot b^x ).
  • Logarithmic form states that if ( y = \log_b(x) ), then ( b^y = x ).
  • Logarithmic properties include:
    • ( \log_b(xy) = \log_b(x) + \log_b(y) )
    • ( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) )
    • ( \log_b(x^n) = n \cdot \log_b(x) ).

• Systems of Equations and Inequalities

  • Methods for solving include graphing, substitution, and elimination.
  • Systems can be linear or nonlinear; solutions to inequalities can be expressed using interval notation.

• Sequences and Series

  • An arithmetic sequence follows ( a_n = a_1 + (n - 1)d ), where ( d ) is the common difference.
  • A geometric sequence is defined by ( a_n = a_1 \cdot r^{(n-1)} ), with ( r ) as the common ratio.
  • Summation formulas:
    • Arithmetic series: ( S_n = \frac{n}{2}(a_1 + a_n) ).
    • Geometric series: ( S_n = a_1 \frac{1 - r^n}{1 - r} ) where ( r \neq 1 ).

• Conic Sections

  • Types of conic sections include circles, ellipses, parabolas, and hyperbolas.
  • Standard equations:
    • Circle: ( (x-h)^2 + (y-k)^2 = r^2 ).
    • Ellipse: ( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 ).
    • Parabola: ( y = a(x-h)^2 + k ).
    • Hyperbola: ( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 ).

• Probability and Statistics

  • Key concepts include mean, median, mode, and range.
  • Important probability rules cover conditional probability, independent events, permutations, and combinations.

• Matrices

  • Defined as rectangular arrays of numbers.
  • Operations include addition, subtraction, multiplication, and finding the determinant.
  • Applications range from solving systems of equations to transformations in geometry.

Important Skills

• Factoring polynomials and solving polynomial equations.
• Understanding functions, including transformations, compositions, and inverses.
• Solving equations that involve exponents and logarithms.
• Analyzing and interpreting various data sets.