Time patterns

Learning Objectives Identify the key characteristics of a periodic function (amplitude, period, midline, phase shift) from a graph representing a time pattern. Model a real-world time-based pattern using a sinusoidal function (sine or cosine). Determine the formula of a periodic function given its graphical representation or key features. Calculate the period of a function from its formula and vice versa. Use a function model to predict future values in a time pattern. Interpret the parameters of a sinusoidal function in the context of a real-world problem. Have you ever noticed how the tides rise and fall, or how a pendulum swings back and forth with perfect regularity? 🌊 These are all functions of time! In this tutorial, we will explore how to use functions, specifically...

TermDefinitionExample Periodic FunctionA function f(t) is periodic if its values repeat over regular intervals. Mathematically, f(t + T) = f(t) for all t, where T is a positive constant.The height of a spot on a spinning wheel over time is a periodic function. If it takes 2 seconds for a full rotation, its height at t=1 second will be the same as at t=3 seconds, t=5 seconds, and so on. Period (T)The smallest positive value of T for which a periodic function repeats. It represents the length of one full cycle, typically measured in units of time.If high tide occurs every 12.4 hours, the period of the tidal function is T = 12.4 hours. Amplitude (A)For a periodic function that oscillates around a central value, the amplitude is the maximum displacement or distance from that central value (th...

General Sinusoidal Function Model f(t) = A \sin(B(t - C)) + D \quad \text{or} \quad f(t) = A \cos(B(t - C)) + D This is the standard form for modeling periodic time patterns. 'A' is the amplitude, 'B' relates to the period, 'C' is the phase shift (horizontal), and 'D' is the midline (vertical shift). Use sine if the pattern starts at the midline and goes up; use cosine if it starts at a maximum. Period-Frequency Relationship T = \frac{2\pi}{B} \quad \text{and} \quad B = \frac{2\pi}{T} This formula connects the period (T) of the function with the parameter 'B' inside the sine or cosine argument. This is crucial for setting up the function to match a real-world time cycle. This formula assumes 't' is measured in a way...