Unit 3A: Trigonometric Functions - Topic 3.1: Periodic Functions

Construct Graphs of Periodic Relationships
- Ability to create graphs based on verbal descriptions provided.
Describe Key Characteristics of a Periodic Function
- Ability to articulate important features of periodic functions.

Periodic:
- A relationship between two variables that repeats in a predictable pattern over successive equal lengths.
Period (of a Function):
- The width of one complete cycle of the function, which corresponds to the distance along the x-axis over which the function starts repeating its values.
Cycle:
- One complete pattern of the function. This term is crucial in understanding how periodic functions behave.
Amplitude:
- Defined as half the distance between the function's maximum and minimum values. Mathematically, if the maximum value is A and the minimum value is B, the amplitude can be calculated as:
  $ext{Amplitude} = \frac{1}{2}(A - B)$
Midline:
- A horizontal line that runs through the middle of the maximum and minimum points of the function. It represents the average value of the function over one cycle.

Periodic Function Definition:
- Consider the function f, which is periodic with a period of 4.
Graphical Representation:
- A portion of the graph of function f is shown, and students are tasked to:
  a) Draw two additional periods for the graph of f off on the axes to the right of the provided graph.
  b) Determine the amplitude. For this example, the amplitude is given as:
  $a = 1$
  c) Identify the midline, which is represented as:
  y = &

Periodic Function g:
- A portion of the graph of the periodic function g is also shown, although specific details about g are not provided in the transcript. Following similar analysis as Example 1, students may be tasked to identify key characteristics such as the period and amplitude based on their observations from the graph.