Problem 8 - Calculus Concepts

Question

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad p(x)=235 \sin (300 x+100)-65\)

Step-by-Step Solution

Answer

Amplitude: 235, Period: \(\frac{\pi}{150}\), Average value: -65, Horizontal shift: \(-\frac{1}{3}\).

Step 1: Determine the Amplitude

The amplitude of a sine function is given by the absolute value of the coefficient of the sine term. For the function \( p(x) = 235 \sin(300x + 100) - 65 \), the amplitude is \(|235|\). Thus, the amplitude is 235.

Step 2: Calculate the Period

The period of a sine function \( y = a \sin(bx + c) + d \) is given by the formula \( \frac{2\pi}{b} \). In this function, \( b = 300 \). Thus, the period is \( \frac{2\pi}{300} \). Simplifying gives the period as \( \frac{\pi}{150} \).

Step 3: Find the Average Value

The average value of the function \( p(x) = a \sin(bx + c) + d \) over one period is the constant term \( d \). In this case, \( d = -65 \). Thus, the average value is -65.

Step 4: Determine the Horizontal Shift

The horizontal shift (also known as the phase shift) of a sine function is determined by solving \( bx + c = 0 \) for \( x \). Here, \( 300x + 100 = 0 \) implies \( x = -\frac{100}{300} = -\frac{1}{3} \). Therefore, the horizontal shift is \(-\frac{1}{3}\) units to the left.

Key Concepts

Amplitude

The amplitude of a trigonometric function measures how far its values stretch above and below a baseline. In the sine function, this baseline is the horizontal axis or the midline of the wave. Simply put, the amplitude dictates how tall or short the peaks and troughs of the wave are.
The formula to find the amplitude is straightforward. It is the absolute value of the coefficient in front of the sine term in the function. For example, in the function \(p(x) = 235 \sin(300x + 100) - 65\), the coefficient of the sine term is 235. Therefore, the amplitude is \(|235|\), which evaluates to 235.
This means the wave extends 235 units above and below its central axis. In practical applications, amplitude could determine the loudness of sound waves or the intensity in many periodic processes.

Periodicity

In trigonometry, the concept of periodicity refers to how often a function repeats its values. For sine and cosine functions, periodicity is determined by the period. The period tells you the length of one complete cycle of the wave.
Calculating the period of a sine function, such as \(y = a \sin(bx + c) + d\), involves using the formula \(\frac{2\pi}{b}\). In our example, \(b = 300\), therefore, the period is \(\frac{2\pi}{300}\). Simplifying this gives \(\frac{\pi}{150}\).
This means the wave completes one full cycle every \(\frac{\pi}{150}\) units along the x-axis. Understanding the period is crucial for applications like signal processing and oscillations, where timing and repetition are central.

Horizontal Shift

Horizontal shift, also known as phase shift, is the movement of a graph along the x-axis. It helps to determine when the waves begin their cycle. To find the horizontal shift in a sine function like \(y = a \sin(bx + c) + d\), you solve the equation \(bx + c = 0\) for \(x\).
For the function \(p(x) = 235 \sin(300x + 100) - 65\), solve \(300x + 100 = 0\) which results in \(x = -\frac{100}{300} = -\frac{1}{3}\). Thus, the graph of the function is shifted \(-\frac{1}{3}\) units to the left.
Understanding horizontal shifts are essential in adjusting the timing of waves. This can be particularly important in fields like engineering and music, where cyclical timing influences outcomes and harmonies.

Average Value

The average value of a periodic function like sine over one complete cycle can often be simplified to a specific term. For the general sine function expressed as \(y = a \sin(bx + c) + d\), the average value is the vertical shift, represented by the constant \(d\).
In the sine equation \(p(x) = 235 \sin(300x + 100) - 65\), the constant term is \(-65\). This signifies that throughout one full cycle of the function, the average of all y-values comes out to be \(-65\).
The average value offers insights into the baseline or mean level of oscillation, crucial for long-term trend observations in disciplines like economics for cycle analysis or meteorology for seasonal patterns.