High School: Functions
Interpreting Functions HSF-IF.A.3
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
When handling functions (careful, they're fragile!), it's easy to see patterns emerge when comparing the x and y values to each other. Students should know that these patterns are not coincidences and, unlike certain patterned wallpaper, they won't make your kitchen look like it belongs in the 1970s.
Students should know that these patterns can be thought of as sequences, or a list of numbers. Sequences can be either arithmetic (where the same number is added or subtracted) or geometric (where the same number is multiplied or divided).
Arithmetic sequences can be converted into functions of the form A(n) = A(1) + (n – 1)d where A(n) is the value of the nth term, A(1) is the value of the first term, n is the term number, and d is the common difference between the terms. So the sequence 3, 8, 13, 18... can be thought of as the function A(n) = 3 + 5(n – 1). That way, the nth term of the sequence will have the value A(n).
Geometric series take a similar form: G(n) = G(1) × rn – 1 where G(n) is the value of the nth term, G(1) is the value of the first term, n is the term number, and r is the common ratio between the terms. So the sequence 3, 6, 12, 24, 48... can be thought of as the function G(n) = 3 × 2n – 1. That way, the nth term of the sequence will have the value G(n).
Students should know that sequences can be defined recursively, or using previous terms to define future terms. For instance, The Fibonacci sequence is a list of numbers where each term is the sum of the two before it. As such, we end up with 1, 1, 2, 3, 5, 8, 13, and so on. We can define this recursively as f(n + 1) = f(n) + f(n – 1). Once the first two terms are defined, (f(0) = f(1) = 1), the sequence can just keep on going, not unlike the Energizer bunny.
Students should be so skilled in seeing the functions in sequences that you should find them accidentally converting other things into functions, too. In fact, it should become second nature.
We're talking analyzing the number of breaths they take as a function of time. The number of times they sneeze as a function of their allergies. The number of detentions they get as a function of how often they kick soccer balls at Coach Gibson. Sorry, Coach.