Problem 225 - Precalculus

Question

For the following exercises, find the amplitude, period, phase shift, and midline. $$ y=\sin \left(\frac{\pi}{6} x+\pi\right)-3 $$

Step-by-Step Solution

Answer

Amplitude = 1, Period = 12, Phase Shift = -6, Midline = y = -3.

Step 1: Identify Amplitude

The amplitude of a sine function in the form of \( y = a \sin{b(x - c)} + d \) is given by \( |a| \). In this equation, the coefficient in front of 'sin' is 1, since \( y = \sin \left( \frac{\pi}{6}x + \pi \right) - 3 \) can be rewritten as \( y = 1 \cdot \sin \left( \frac{\pi}{6}x + \pi \right) - 3 \). Therefore, the amplitude is \( 1 \).

Step 2: Calculate the Period

The period of a sine function is calculated using \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function. Here, \( b = \frac{\pi}{6} \). Thus, the period is \( \frac{2\pi}{\frac{\pi}{6}} = 12 \).

Step 3: Find the Phase Shift

The phase shift of a sine function \( y = a \sin{b(x - c)} + d \) is given by \( c \), where \( bx + bc \) is set equal to zero to find \( c \). Here we have \( \frac{\pi}{6}x + \pi = 0 \). Solving for \( x \), we have \( \frac{\pi}{6}x = -\pi \), leading to \( x = -6 \), so the phase shift is \(-6\).

Step 4: Determine the Midline

The midline of a sine function is given by \( y = d \), where \( d \) is the vertical shift. In this equation, \( d = -3 \), so the midline is \( y = -3 \).

Key Concepts

Amplitude

Amplitude refers to the height of the wave created by a trigonometric function from its central position, with this central position often being referred to as the "midline." It's crucial to understand that amplitude is always a positive number, representing the absolute value of the coefficient in front of the sine function. For our function, which can be written as \[y = 1 imes \sin \left( \frac{\pi}{6} x + \pi \right) - 3\]we see that the amplitude is determined by the coefficient 1. Thus, the amplitude is 1. This means the wave oscillates 1 unit above and below its midline.

Period

The period of a trigonometric function signifies how long it takes for the wave to complete one full cycle before it repeats itself. For sine and cosine functions, the default period is \(2\pi\), but it can be modified by the coefficient \(b\) in the function's argument \(b(x - c)\). The formula to find the period is \[\frac{2\pi}{b}\]In our function, the coefficient \(b\) is \(\frac{\pi}{6}\), so to find the period we calculate \[\frac{2\pi}{\frac{\pi}{6}} = 12\]This tells us the wave will repeat every 12 units along the x-axis.

Phase shift

Phase shift determines how much a graph is horizontally shifted from its conventional position. Addressed by \(c\) in the equation \(b(x - c)\), the phase shift shows how far and in which direction the graph is moved. By setting the equation \[\frac{\pi}{6}x + \pi = 0\]we solve for \(x\): \[\frac{\pi}{6}x = -\pi\]yielding \[x = -6\]Hence, the entire graph is shifted 6 units to the left. Phase shifts can sometimes confuse students, but remembering to set the inner argument equal to zero can simplify the process.

Midline

The midline is essentially a horizontal line that represents the average value or the central axis of the trigonometric function. It is influenced by the vertical shift \(d\) in the function's equation \[y = a \cdot \sin(b(x - c)) + d\]In our specific function, \(d = -3\). This means that the midline, or the central horizontal line around which the sine function oscillates, is at \(y = -3\). Understanding the midline helps provide clarity on how high and how low the wave moves in relation to a fixed horizontal reference point.