Problem 226 - Precalculus

Question

For the following exercises, find the amplitude, period, phase shift, and midline. $$ y=8 \sin \left(\frac{7 \pi}{6} x+\frac{7 \pi}{2}\right)+6 $$

Step-by-Step Solution

Answer

Amplitude: 8; Period: \(\frac{12}{7}\); Phase shift: -3; Midline: \(y = 6\).

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function in the form \(y = a \sin(bx + c) + d\) is given by the absolute value of \(a\). In this function, \(a = 8\), so the amplitude is \(8\).

Step 2: Determine the Period

The period of a sinusoidal function \(y = a \sin(bx + c) + d\) is calculated using the formula \(\frac{2\pi}{|b|}\). Here, \(b = \frac{7\pi}{6}\). Thus, the period is \(\frac{2\pi}{|\frac{7\pi}{6}|} = \frac{12}{7}\).

Step 3: Calculate the Phase Shift

The phase shift is found by solving \(-\frac{c}{b}\) for the equation \(y = a \sin(bx + c) + d\). Here, \(c = \frac{7\pi}{2}\) and \(b = \frac{7\pi}{6}\). Plug these into the formula to get the phase shift: \(-\frac{\frac{7\pi}{2}}{\frac{7\pi}{6}} = -3\).

Step 4: Identify the Midline

The midline of the function \(y = a \sin(bx + c) + d\) is determined by the value of \(d\). Here, \(d = 6\), so the midline is \(y = 6\).

Key Concepts

Amplitude

In trigonometric functions, the amplitude refers to the height of the wave from the center line to its peak. It measures the maximum distance the function reaches from its midline. To find the amplitude of a sine or cosine function given in the form \( y = a \sin(bx + c) + d \), we look at the absolute value of the coefficient \( a \).
In the example, the function is \( y = 8 \sin \left(\frac{7 \pi}{6} x + \frac{7 \pi}{2}\right) + 6 \). Since \( a = 8 \), the amplitude is \( |8| = 8 \).

• The amplitude captures how high or low the wave gets compared to its rest position, which means an amplitude of 8 indicates a fluctuation 8 units both above and below the midline.
• Understanding amplitude helps predict the maximum extent of the wave, crucial in scenarios like sound waves, where amplitude might correlate with volume.

Period

The period of a trigonometric function tells us how long it takes for the wave to complete one full cycle. For sine and cosine functions, the standard period is \( 2\pi \). However, this period can be altered by the coefficient \( b \) in the expression \( y = a \sin(bx + c) + d \).
The formula to calculate the period is \( \frac{2\pi}{|b|} \).
In our exercise, the function is \( y = 8 \sin \left(\frac{7 \pi}{6} x + \frac{7 \pi}{2}\right) + 6 \), and \( b = \frac{7\pi}{6} \). Plugging \( b \) into the formula gives:
\[ \frac{2\pi}{\left| \frac{7\pi}{6} \right|} = \frac{12}{7} \]

• This indicates the wave completes one cycle over a horizontal length of \( \frac{12}{7} \).
• A smaller value of \( \frac{12}{7} \) compared to \( 2\pi \) in the basic sine function means waves happen more frequently.
• Period calculation is critical when analyzing cycles, such as in time-based phenomena like daily temperature fluctuations.

Phase shift

Phase shift refers to the horizontal movement of a trigonometric function from its usual position. In the equation \( y = a \sin(bx + c) + d \), phase shift can be calculated with the formula \(-\frac{c}{b}\).
This tells us how far and in which direction the wave has been shifted along the x-axis.
For the function \( y = 8 \sin \left(\frac{7 \pi}{6} x + \frac{7 \pi}{2}\right) + 6 \), with \( c = \frac{7\pi}{2} \) and \( b = \frac{7\pi}{6} \), the phase shift is:
\[ -\frac{\frac{7\pi}{2}}{\frac{7\pi}{6}} = -3 \]

• This negative result tells us the graph is shifted 3 units to the left.
• The phase shift is critical for aligning wave functions with real-world data, such as predicting high tides based on known cycles.
• Recognizing phase shift helps in understanding where the function starts its cycle.

Midline

The midline of a trigonometric function is the horizontal line that runs through the middle of the wave, representing the average value of the maximum and minimum points of the function. In the formula \( y = a \sin(bx + c) + d \), the midline corresponds to the constant term \( d \).
For the given function \( y = 8 \sin \left(\frac{7 \pi}{6} x + \frac{7 \pi}{2}\right) + 6 \), \( d = 6 \), so the midline is \( y = 6 \).

• The midline impacts how high or low the entire graph can shift vertically.
• Understanding the midline is essential for interpreting the general position of the wave concerning its environment.
• For example, in alternating current electricity, the midline would represent zero voltage.