Graph the equation y=f(i) on the interval [0,24] . Let y...

Graph the equation $y=f(i)$ on the interval $[0,24] .$ Let $y$ represent the outdoor temperature (in "F) at time $t$ (in hours), where $t=0$ corresponds to 9 A.M. Describe the temperature during the 24 -hour interval.
$$y=80+22 \cos \left[\frac{\pi}{12}(t-3)\right]$$

Key Concepts

Vertical Translation

Vertical translation refers to the shifting of a graph up or down on the vertical axis. This moves the midline of a periodic function to a new position without altering its shape. In temperature models, vertical translation adjusts the baseline temperature around which the variation occurs.

Phase Shift

A phase shift is the horizontal translation of a function's graph. It determines where the cycle of the periodic function begins relative to the standard position. Understanding the phase shift helps in aligning the graph with real-world timing, indicating the time at which a particular event (like the peak temperature) occurs.

Amplitude

The amplitude of a function is the distance from the midline to the maximum (or minimum) value. It represents the magnitude of variation in a periodic function. In applications like temperature modeling, the amplitude indicates how much the temperature deviates from the average value.

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or periods. They are foundational in modeling phenomena that occur cyclically, such as daily temperature variations, seasons, and sound waves. In this context, understanding a periodic function allows us to predict and analyze behavior over time.

Period

The period of a periodic function is the length of the interval over which the function completes one full cycle before repeating. This concept is crucial for determining the frequency of occurrence of the repeating pattern. In a daily temperature cycle, the period describes the duration of one complete cycle of temperature change.

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