Scientists sometimes use the formula f(b)=a sin(b t+c)+d to...

Scientists sometimes use the formula $$f(b)=a \sin (b t+c)+d$$ to simulate temperature variations during the day, with time $t$ in hours, temperature $f(t)$ in $^{\circ} \mathbf{C},$ and $t=\mathbf{0}$ corresponding to midnight. Assume that $f(t)$ is decreasing at midnight. (a) Determine values of $a, b, c,$ and $d$ that fit the information. (b) Sketch the graph of $f$ for $0 \leq t \leq 24$ The high temperature is $10^{\circ} \mathrm{C},$ and the low temperature of $-10^{\circ} \mathrm{C}$ occurs at $4 \mathrm{A} . \mathrm{M}$

Scientists sometimes use the formula
$$f(b)=a \sin (b t+c)+d$$
to simulate temperature variations during the day, with time $t$ in hours, temperature $f(t)$ in $^{\circ} \mathbf{C},$ and $t=\mathbf{0}$ corresponding to midnight. Assume that $f(t)$ is decreasing at midnight.
(a) Determine values of $a, b, c,$ and $d$ that fit the information.
(b) Sketch the graph of $f$ for $0 \leq t \leq 24$
The high temperature is $10^{\circ} \mathrm{C},$ and the low temperature of $-10^{\circ} \mathrm{C}$ occurs at $4 \mathrm{A} . \mathrm{M}$

Key Concepts

Derivative and Monotonicity

The derivative of a sinusoidal function is used to analyze its increasing or decreasing behavior at specific points. Understanding the sign of the derivative is crucial for ensuring that the modeled process (such as temperature change) behaves realistically at given times, reflecting trends like warming or cooling.

Phase Shift

Phase shift is the horizontal translation of the sinusoidal graph, controlled by the additive constant inside the sine function's argument. It determines the starting point of the cycle and aligns key features of the function, such as peaks or troughs, with specific moments in time.

Vertical Translation

Vertical translation shifts the sinusoidal function up or down on the graph and is determined by the constant added to the function. It represents the midline of the oscillation, indicating the average value around which the periodic variations occur.

Amplitude

Amplitude represents the magnitude of the oscillations in a sinusoidal function. It is the coefficient in front of the sine (or cosine) term and determines how far the function values deviate from the midline, reflecting the maximum increase and decrease relative to the average value.

Sinusoidal Functions

Sinusoidal functions are mathematical models used to represent periodic phenomena. They are characterized by repeating patterns described using sine or cosine functions, and they are particularly useful in simulating natural processes that oscillate over time.

Period

The period of a sinusoidal function is the length of one complete cycle of the oscillation. Determined by the parameter that multiplies the variable inside the sine function, it defines how rapidly the cycles repeat and is given by the formula 2? divided by that parameter.

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