Times of everyday events

Learning Objectives Model periodic daily events, such as daylight hours or tides, using sinusoidal functions. Determine the amplitude, period, phase shift, and vertical shift of an event from a given data set or graph. Construct a mathematical equation to represent the timing of a cyclical event. Use a sinusoidal model to predict the value of a variable at a specific time. Model events occurring at regular intervals using arithmetic sequences. Calculate the time of the nth occurrence of an event described by an arithmetic sequence. Ever wondered how meteorologists can predict the exact time of sunrise for any day of the year, or how a smartphone app knows the time of the next high tide? ☀️🌊 In this lesson, we will explore how the advanced mathematical tools you're lea...

TermDefinitionExample Periodic FunctionA function that repeats its values at regular intervals or periods. The patterns of many everyday events, like the swing of a pendulum or the daily temperature cycle, can be described by periodic functions.The function describing the height of a point on a spinning Ferris wheel over time is periodic, as it repeats with every full rotation. Sinusoidal ModelA mathematical model using a sine or cosine function to represent periodic data. It is defined by its amplitude, period, phase shift, and vertical shift.The number of daylight hours in a non-equatorial city over a year can be closely approximated by a sinusoidal model, peaking in summer and hitting a minimum in winter. Amplitude (A)For a periodic event, the amplitude is half the distance between the...

General Sinusoidal Model y = A \cos(B(t - C)) + D This formula models a periodic event over time 't'. 'A' is the amplitude, 'D' is the vertical shift (average value), 'C' is the phase shift (start time of the cycle), and 'B' is related to the period. We often use cosine because its cycle naturally starts at a maximum, which is easy to identify in many real-world events (e.g., longest day of the year, highest tide). Period Calculation \text{Period} = \frac{2\pi}{|B|} This formula connects the period of the event (how long one cycle takes) to the 'B' value in the sinusoidal model. When building a model, you often know the period and must solve for B using B = 2π / Period. Arithmetic Sequence Term Formula t_n = t...