Glossary
Arithmetic Sequences
A sequence of numbers where the difference between consecutive terms is constant.
Example:
If you save an additional 10, your savings over time form an arithmetic sequence: 15, 25...
Common difference
The constant value added to each term in an arithmetic sequence to get the next term.
Example:
In the sequence 2, 5, 8, 11..., the common difference is 3, as each term is 3 more than the previous one.
Common ratio
The constant factor by which each term in a geometric sequence is multiplied to get the next term.
Example:
In the sequence 100, 50, 25, 12.5..., the common ratio is 0.5, indicating a halving of the value each time.
Composite Functions
A function formed by applying one function to the result of another function.
Example:
If f(x) calculates the cost of a shirt and g(x) calculates a discount, then (f ∘ g)(x) would represent applying the discount first, then finding the cost of the discounted shirt, making it a composite function.
Exponential functions
Functions of the form f(x) = ab^x that model situations where a quantity changes by a constant multiplicative factor over equal intervals.
Example:
The amount of caffeine remaining in your body after drinking coffee can be modeled by an exponential function, as it decreases by a certain percentage each hour.
Geometric Sequences
A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
Example:
A chain letter where each person sends it to 3 new people creates a geometric sequence of recipients: 1, 3, 9, 27...
Inverse Functions
Two functions are inverses if one function 'undoes' the operation of the other, meaning their composition results in the original input.
Example:
If a function converts Celsius to Fahrenheit, its inverse function would convert Fahrenheit back to Celsius.
Logarithmic functions
Functions of the form f(x) = log_b(x) that are the inverse of exponential functions, used to determine the exponent to which a base must be raised to produce a given number.
Example:
The Richter scale uses a logarithmic function to measure earthquake intensity, where each whole number increase represents a tenfold increase in amplitude.
Semi-log Plots
A graph where one axis uses a logarithmic scale and the other uses a linear scale, often used to visualize exponential relationships as straight lines.
Example:
When tracking the rapid growth of a viral video's views, a semi-log plot can make the exponential increase appear as a straight line, simplifying analysis of the growth rate.