Sequences and Functions

Introduction to Sequences

Sequences are ordered lists of numbers that follow a specific pattern or rule. They are foundational in algebra for understanding patterns, functions, and mathematical modeling.

• Arithmetic Sequence: Each term is obtained by adding a constant difference to the previous term.

• Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio.

• Recursive Formula: Defines each term based on the previous term(s).

• Explicit Formula: Defines the nth term as a function of n.

• Sequences as Functions: Sequences can be represented as functions with domain as positive integers.

Example: The sequence 2, 4, 6, 8, ... is arithmetic with a common difference of 2.

Polynomial Functions

Understanding Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

• Polynomial Vocabulary: Terms, degree, leading coefficient, constant term.

• Operations: Addition, subtraction, and multiplication of polynomials.

• Factoring: Expressing a polynomial as a product of its factors.

• Zeros and Factors: The zeros of a polynomial are the values of x for which the polynomial equals zero.

• End Behavior: The behavior of the graph as x approaches positive or negative infinity.

• Polynomial Division: Long division and synthetic division methods.

Example: factors to .

Rational Expressions and Functions

Rational Functions and Equations

Rational expressions are ratios of polynomials. Rational functions are functions defined by rational expressions.

• Asymptotes: Lines that the graph approaches but never touches.

• Domain and Range: The set of possible input and output values.

• Solving Rational Equations: Involves finding common denominators and checking for extraneous solutions.

• Polynomial Identities: Equations that are true for all values of the variables involved.

Example: is undefined at (vertical asymptote).

Exponents, Radicals, and Complex Numbers

Exponent Properties and Radical Equations

Exponents and radicals are used to represent repeated multiplication and roots, respectively. Complex numbers extend the real numbers to include solutions to equations like .

• Exponent Rules: Product, quotient, power, zero, and negative exponent rules.

• Rational Exponents:

• Complex Numbers: Numbers of the form , where .

• Quadratic Equations: Can have real or complex solutions.

• Quadratic Formula:

Example:

Exponential and Logarithmic Functions

Growth, Decay, and Logarithms

Exponential functions model rapid growth or decay. Logarithms are the inverses of exponential functions and are used to solve equations involving exponents.

• Exponential Growth/Decay: where (growth), (decay).

• Logarithmic Functions: is the inverse of .

• Natural Logarithm: is the logarithm with base .

• Solving Exponential Equations: Use logarithms to solve for the exponent.

Example: because .

Transformations of Functions

Translations, Reflections, and Scaling

Transformations change the position or shape of a function's graph. Common transformations include translations (shifts), reflections, and dilations (scaling).

• Translation: Shifting the graph horizontally or vertically.

• Reflection: Flipping the graph over a line (e.g., x-axis or y-axis).

• Scaling: Stretching or compressing the graph.

• Even and Odd Functions: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

Example: is the graph of shifted right by 2 units.

Additional Topics

Sequences, Series, and the Binomial Theorem

Sequences and series involve ordered lists and their sums. The Binomial Theorem provides a formula for expanding powers of binomials.

• Arithmetic Series: Sum of an arithmetic sequence.

• Geometric Series: Sum of a geometric sequence.

• Binomial Theorem:

Example: The sum of the first n natural numbers is .

Statistical Inferences

Distributions and Experimental Data

Statistical inference involves drawing conclusions about populations based on samples. Key concepts include distributions, standard deviation, normal distribution, and confidence intervals.

• Standard Deviation: Measures the spread of data around the mean.

• Normal Distribution: A bell-shaped curve describing many natural phenomena.

• Confidence Interval: A range of values likely to contain the population parameter.

Example: In a normal distribution, about 68% of data falls within one standard deviation of the mean.