Low temperature in Fairbanics Based on years of weather...

Low temperature in Fairbanics Based on years of weather data, the expected low temperature $T$ (in 'F) in Fairbanks, Alaska, can be approximated by $T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14$ where $t$ is in days and $t=0$ corresponds to January I. A. Sketch the graph of $T$ for $0 \leq t \leq 365$ B. Predict when the coldest day of the year will occur.

Low temperature in Fairbanics Based on years of weather data, the expected low temperature $T$ (in 'F) in Fairbanks, Alaska, can be approximated by
$T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14$
where $t$ is in days and $t=0$ corresponds to January I.
A. Sketch the graph of $T$ for $0 \leq t \leq 365$
B. Predict when the coldest day of the year will occur.

Key Concepts

Graphing Transformed Trigonometric Functions

Graphing transformed trigonometric functions involves applying modifications such as amplitude, period, phase shift, and vertical shift to the basic sine or cosine graph. This process helps in accurately representing real-world cyclical data by aligning the graph with the actual peaks, troughs, and baseline values observed in the data.

Critical Points

Critical points, such as the maximum and minimum values of a function, are essential in analyzing a function's behavior. For a sine function modeling temperature, identifying the minima (or maxima, depending on context) allows for the prediction of important events—like the coldest day of the year. Recognizing these points aids in interpreting and applying the model to real-life scenarios.

Vertical Shift

A vertical shift moves the entire graph of a function up or down on the coordinate plane. In the context of a sine function, this shift represents the average or baseline value around which the oscillations occur. The vertical shift adjusts the central value of the function to better fit the data being modeled.

Phase Shift

The phase shift in a trigonometric function represents a horizontal translation of the graph along the x-axis. It indicates where in its cycle the function begins relative to a standard sine function. Understanding the phase shift allows one to determine the timing of key features in the cycle, such as the position of peaks or troughs.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. They are fundamental in modeling phenomena that cycle over time, such as temperatures, sound waves, and seasonal variations. Recognizing the repeating nature of these functions helps in analyzing patterns and forecasting future behavior based on past cycles.

Amplitude

Amplitude refers to the maximum deviation of the function from its midline or equilibrium position. In the context of a sine function, it is the distance from the center line to the peak or trough of the wave. The amplitude determines the magnitude of variation in the modeled phenomenon, such as the range of temperature fluctuation.

Sine Function

The sine function is a trigonometric function that naturally models smooth, periodic oscillations. It is commonly used to represent phenomena with periodic behavior due to its cyclic nature. Key characteristics of the sine function include its amplitude, period, phase shift, and vertical shift, all of which affect the shape and position of its graph.

Period

The period of a function is the length of the interval over which the function completes one full cycle before repeating. For a sine function, the period can be calculated as 2? divided by the coefficient of the variable inside the sine function. The period is crucial in understanding the frequency and timing of the repetitive behavior of the function.

Please give Ace some feedback