2.1 Arithmetic and Geometric Sequences

• sequence: function from the whole numbers (n) to real numbers

• graph consists of discrete points instead of curves because we are dealing with whole numbers

Arithmetic

• sequence of numbers with a common difference

• (n)th term is given by a^1 + (n-1)d

• d = the common difference CONSTANT RATE OF CHANGE

• input = term number

• output = term value

• can be positive (increasing) or negative (decreasing)

Geometric

• sequence of numbers with a common ratio

• (n)th term is given by g^1 ( r )^(n-1)

• r = common ratio PROPORTIONAL CHANGE

• can be positive (increasing) or negative (decreasing)

• decreasing = r < 1 / Increasing = r > 1

2.2 Change in Linear & Exponential Functions

Linear Functions (arithmetic)

• f(x)= mx + b

• the first term is represented by a^0

• the roc is represented by d or “m”

• must have a constant roc

• point-slope formula and explicit formula are NOT the same

Exponential Functions (geometric)

• f(x)= a(b)^x

• the ratio of consecutive terms is the same

• represented by g^n = g^0 ( r )^n

• functions DO NOT = sequences

• they may have different domains and ranges

2.3 Exponential Functions

• x is always in the exponent

• 0 < x < 1 = exponential decay (concave down)

• x > 1 = exponential growth (concave up)

• domain always = all real numbers

• output values are proportional over equal-length consecutive input values

• ALWAYS decreasing or increasing

• NO points of inflection or extrema

• g(x) = f(x) + k is an additive transformation (up by k units)

• make sure to know the limit statements

2.4 Exponential Function Manipulation

Properties

• power to power means multiplying powers

• multiplying means adding powers

• negative exponent property means x^-n = 1/x^n *ONLY VALID FOR POSITIVE BASES*

• exponential unit fraction- b^1/k ( k= natural numbers)

• rational exponents- b^1/n = n√b

2.5 Exponential Function Context & Data Modeling

Input - Output Pairs

• can construct exponential functions

• initial value = a / base= b

• set up a system of equations ( or through an exponential regression) to find the best-fit function

• correlation coefficient = r²: measures how well the data fits

• residuals can help determine if the graph is appropriate

• f(x) = e^x (continuous growth)

• f(x) = e^-x (continuous decay)

2.6 Competing Function Model Validation

• identify patterns

• remember to consider the domain, range, and purpose of the function

Appropriation

• analyze residuals of regressions

• no pattern means it is appropriate

• predicted trends won’t always fit points (errors will happen)

• Errors are the predicted value minus the actual value

2.7 Composite Functions

• made up of 2 or more simpler functions put together

• combining functions = composition of functions

• basic notation: f(g(x)) aka “f of g of x”

• - output of the inside function is the input of the second function

• 1. identify inside and outside function

  2. substitute each x with the inner function

2.8 Inverse Functions

Criteria

• the function must be a 1 to 1 ratio

• the domain must not be restricting

• notation = f^-1

• domain and range will swap in an inverse

• NOT ALL FUNCTIONS HAVE AN INVERSE

• INVERSES ARE NOT ALWAYS FUNCTIONS

Steps to Finding the Inverse

• change f(x) to y

• swap x and y roles

• find inverse by solving for y