Algebra 1: Functions and Sequences

• A function in algebra 1 is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• Functions can be represented in various ways, including equations, graphs, tables, and verbal descriptions

Integer Sequences and Patterns

• An integer sequence is an ordered list of integers
• Identifying patterns in integer sequences is a common mathematical exercise, but one should be cautious about assuming a pattern will continue indefinitely
• Recognizing a pattern and writing a formula is useful; the formula should be tested to make sure it works
• Given a few initial terms, multiple formulas can often fit the observed pattern

Recursive Formulas for Sequences

• A recursive formula defines a term in a sequence based on the preceding terms
• To define a sequence recursively, you need to specify the initial term(s) and a rule for finding subsequent terms

Arithmetic Sequences

• An arithmetic sequence is a sequence where the difference between consecutive terms is constant
• This constant difference is called the common difference, denoted as 'd'
• The general form of an arithmetic sequence is: a, a + d, a + 2d, a + 3d, ... , where 'a' is the first term
• The n-th term (an) of an arithmetic sequence can be expressed as: an = a + (n - 1)d

Geometric Sequences

• A geometric sequence is a sequence where the ratio between consecutive terms is constant
• This constant ratio is called the common ratio, denoted as 'r'
• The general form of a geometric sequence is: a, ar, ar^2, ar^3, ..., where 'a' is the first term
• The n-th term (an) of a geometric sequence can be expressed as: an = a * r^(n-1)

Recursive Formulas for Arithmetic Sequences

• A recursive formula expresses each term in relation to the preceding term(s)
• For an arithmetic sequence, the recursive formula is: a1 = a, an = an-1 + d, where n > 1, 'a' is the first term, and 'd' is the common difference
• This means to find any term, you add the common difference to the previous term
• Example: If the arithmetic sequence is 2, 5, 8, 11,..., the recursive formula is a1 = 2, an = an-1 + 3

Recursive Formulas for Geometric Sequences

• For a geometric sequence, the recursive formula is: a1 = a, an = an-1 * r, where n > 1, 'a' is the first term, and 'r' is the common ratio
• Each term is found by multiplying the previous term by the common ratio
• Example: If the geometric sequence is 3, 6, 12, 24,..., the recursive formula is a1 = 3, an = an-1 * 2

Examples of Arithmetic Sequence Applications

• Suppose you start with $100 and add $20 each week to your savings
• The sequence of savings each week would be: 100, 120, 140, 160, ...
• The first term is 100 and the common difference is 20
• The n-th term is given by: an = 100 + (n - 1) * 20
• After 10 weeks, the amount saved would be a10 = 100 + (10 - 1) * 20 = $280

Examples of Geometric Sequence Applications

• Consider a population of bacteria that doubles every hour, starting with 5 bacteria
• The sequence of the number of bacteria each hour would be: 5, 10, 20, 40,...
• The first term is 5 and the common ratio is 2
• The n-th term is given by: an = 5 * 2^(n-1)
• After 6 hours, the number of bacteria would be a6 = 5 * 2^(6-1) = 5 * 32 = 160

Cautions when Predicting with Patterns

• With limited terms, one can find multiple formulas that agree with the initial terms, but diverge later on
• It's important to seek more data or context to validate the proposed pattern or formula
• Patterns observed in real-world data may change due to external factors or underlying processes not immediately apparent
• Extrapolation, i.e. predicting beyond the known data, should be done cautiously

Functions

• Functions take an input, process it, and generate an output
• Every input has one and only one output

Domain and Range

• The domain of a function represents all possible input values
• The range of a function represents all possible output values

Function Notation

• Functions are often written as f(x), where x is the input
• f(x) represents the output of the function for a given input x

Linear Functions

• Linear functions have a constant rate of change and can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept

Exponential Functions

• Exponential functions involve a constant base raised to a variable exponent, often used to model growth or decay

Quadratic Functions

• Quadratic functions have the form f(x) = ax^2 + bx + c, where a, b, and c are constants
• These functions form parabolas when graphed