Problem 174 - Precalculus

Question

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes. $$ f(x)=6 \sin \left(3 x-\frac{\pi}{6}\right)-1 $$

Step-by-Step Solution

Answer

Amplitude: 6, Period: \(\frac{2\pi}{3}\), Midline: \(y = -1\), No asymptotes.

Step 1: Identify Amplitude

The function given is \( f(x) = a \sin(bx - c) + d \) where \( a = 6 \). The amplitude of a sinusoidal function is given by its coefficient \( |a| \). Thus, the amplitude is \( |6| = 6 \).

Step 2: Find the Period

The period of a sinusoidal function \( \sin(bx - c) \) is given by \( \frac{2\pi}{|b|} \). We have \( b = 3 \), so the period is \( \frac{2\pi}{3} \). This means each full wave of the sine function spans \( \frac{2\pi}{3} \) units along the x-axis.

Step 3: Calculate the Midline

The midline of the function is given by the constant \( d \) in the equation \( f(x) = a \sin(bx - c) + d \). Here, \( d = -1 \), so the equation of the midline is \( y = -1 \).

Step 4: Determine the Phase Shift

Phase shift is calculated using \( \frac{c}{b} \) where \( c = \frac{\pi}{6} \) and \( b = 3 \). Thus, the phase shift is \( \frac{\pi/6}{3} = \frac{\pi}{18} \). This means the graph is shifted \( \frac{\pi}{18} \) units to the right.

Step 5: Identify Asymptotes (If Applicable)

Since this is a sine function, there are no vertical asymptotes. This property would apply if the function were tangent or cotangent.

Key Concepts

Amplitude

In trigonometric functions, amplitude refers to the height of the wave from its midline. It determines how "tall" or "short" the function appears on a graph. For sine and cosine functions, amplitude is calculated by the absolute value of the leading coefficient.

Given the function \( f(x) = 6 \sin(3x - \frac{\pi}{6}) - 1 \), we identify the amplitude as \( |6| = 6 \). This means the graph of the sine function reaches 6 units above and below its midline.

• Amplitude tells us the stretch or compression of the sinusoidal function vertically.
• The larger the amplitude, the higher the peaks and lower the troughs.

Period

The period of a trigonometric function defines the length of one complete wave cycle along the x-axis. It is calculated for sine and cosine functions using the formula \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \).

In our function, \( b = 3 \), so the period is \( \frac{2\pi}{3} \). This means the sine wave repeats itself every \( \frac{2\pi}{3} \) units.

• A shorter period indicates that the waves are closer together.
• A longer period indicates wider-spaced waves.

Midline Equation

The midline of a trigonometric function is a horizontal line that acts as the center or baseline of its wave. It is represented by the constant \( d \) in the equation \( f(x) = a \sin(bx - c) + d \).

For the function \( f(x) = 6 \sin(3x - \frac{\pi}{6}) - 1 \), the midline is \( y = -1 \). The function's maximum and minimum values oscillate equally above and below this line.

• The midline shifts up or down depending on the value of \( d \).
• It acts as an axis of symmetry for the wave.

Phase Shift

Phase shift in trigonometric functions describes the horizontal shift of the wave on the graph. It determines where the wave starts compared to the typical position. The phase shift is calculated as \( \frac{c}{b} \), where \( c \) is a constant from the function \( ax \sin(bx - c) + d \).

For this function, with \( c = \frac{\pi}{6} \) and \( b = 3 \), the phase shift is \( \frac{\pi/6}{3} = \frac{\pi}{18} \), indicating the wave moves \( \frac{\pi}{18} \) units to the right.

• A positive phase shift moves the wave right.
• A negative phase shift moves the wave left.

Asymptotes

In trigonometric functions like sine and cosine, there are no vertical asymptotes. Asymptotes are lines the graph approaches but never touches. They are crucial for understanding functions like tangent and cotangent that have undefined points.

For this sine function, \( f(x) = 6 \sin(3x - \frac{\pi}{6}) - 1 \), asymptotes are not present. These functions are continuous and smooth with no breaks.

• Sine and cosine functions are defined for all real numbers.
• If we were dealing with tangent or cotangent functions, we'd search for points where the graph becomes infinitely large or small.