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.