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How do you find the period of a function from its graph?
Identify a repeating pattern, then measure the length of one complete cycle (from start to the end of the repeating pattern).
If f(x) is periodic with period P, and you know f(a) = b, what other values of x will also give f(x) = b?
f(a + P) = b, f(a + 2P) = b, f(a + nP) = b, where n is any integer.
Define a periodic function.
A function that repeats its values in regular intervals or cycles.
What is a cycle in the context of periodic functions?
One complete repetition of the pattern in a periodic function.
Define the period of a periodic function.
The length of one complete cycle; the smallest interval over which the function repeats.
Explain how to identify a periodic relationship.
Look for repeating patterns in the output values as the input values increase at regular intervals.
What characteristics remain consistent across all periods of a periodic function?
Intervals of increase and decrease, concavity, and rates of change.
How can the graph of a single cycle be used to understand the entire periodic function?
Because the pattern is consistent across all cycles, the graph of a single cycle contains all the information needed to understand the function's behavior.