Problem 18 - A Graphical Approach to Precalculus with Limits

Question

March each function defined in Column I with the appropriate description in Column \(\mathrm{II.}\) Do not use a calculator. I $$y=2 \sin (3 x-4)$$ II A. Amplitude \(=2 ;\) period \(=\frac{\pi}{2} ;\) phase shift \(=\frac{3}{4}\) B. Amplitude \(=3 ;\) period \(=\pi ;\) phase shift \(=2\) C. Amplitude \(=4 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{2}{3}\) D. Amplitude \(=2 ;\) period \(=\frac{2 \pi}{3} ;\) phase shift \(=\frac{4}{3}\)

Step-by-Step Solution

Answer

Match with D: Amplitude = 2; Period = \(\frac{2\pi}{3}\); Phase Shift = \(\frac{4}{3}\).

Step 1: Identify Components of the Function

The function is given by \( y = 2 \sin (3x - 4) \). The general form for a sine function is \( y = A \sin(Bx - C) \), where \( A \) is the amplitude, the period is calculated as \( \frac{2\pi}{B} \), and the phase shift is obtained by \( \frac{C}{B} \).

Step 2: Calculate Amplitude

The amplitude \( A \) is the coefficient in front of the sine function. For the function \( y = 2 \sin (3x - 4) \), \( A = 2 \). This means the amplitude is 2.

Step 3: Calculate Period

The period of a sine function \( y = A \sin(Bx - C) \) is given by \( \frac{2\pi}{B} \). Here, \( B = 3 \), so the period is \( \frac{2\pi}{3} \).

Step 4: Calculate Phase Shift

The phase shift of the sine function is given by \( \frac{C}{B} \). For the function \( y = 2 \sin(3x - 4) \), \( C = 4 \) and \( B = 3 \). Therefore, the phase shift is \( \frac{4}{3} \).

Step 5: Match with Column II

From the calculations: Amplitude \( = 2 \), Period \( = \frac{2\pi}{3} \), Phase Shift \( = \frac{4}{3} \). This matches description D from Column II.

Key Concepts

Amplitude

In the realm of trigonometric functions, the amplitude of the sine function is a crucial concept. Amplitude refers to the height from the centerline of the wave to its peak. When we talk about the function \( y = A \sin(Bx - C) \), the amplitude is represented by \( A \). This value determines how tall or short the wave is. For instance, in the function \( y = 2 \sin(3x - 4) \), the amplitude is \( 2 \).

• The amplitude is always a positive number, even if it's expressed as a negative in the equation, since it measures a "distance."
• An amplitude of \( 2 \) means the wave oscillates 2 units above and below the central axis.

Understanding amplitude helps in visualizing the vertical stretch or shrink of a trigonometric graph. It indicates the strength or intensity of the oscillation. Remember, the greater the amplitude, the taller the wave.

Period of Sine Functions

The period of a sine function is the length of one complete cycle of the wave, or the distance after which the pattern repeats. It is a key factor in determining how "stretched" out the wave appears horizontally. For the sine function \( y = A \sin(Bx - C) \), the period is calculated as \( \frac{2\pi}{B} \).In the function \( y = 2 \sin(3x - 4) \), \( B \) is \( 3 \). Thus, the period is \( \frac{2\pi}{3} \). This means that every \( \frac{2\pi}{3} \) units, the wave begins to repeat its shape.

• The period tells us how quickly a sine curve returns to its starting point.
• An important property of the period is that it can never be zero, as a wave must be continuous.

Knowing the period aids in drawing the accurate graph, as it determines the width of one sine wave cycle along the x-axis.

Phase Shift

Phase shift refers to the horizontal movement of the graph of a sine function along the x-axis. It's essentially a slide to the left or right. In our sine function model \( y = A \sin(Bx - C) \), the phase shift is determined by \( \frac{C}{B} \). It's the value of \( C \) divided by \( B \).For the function \( y = 2 \sin(3x - 4) \), the phase shift is \( \frac{4}{3} \). This value tells us that the sine wave is shifted \( \frac{4}{3} \) units to the right along the x-axis.

• A positive phase shift indicates a shift to the right, while a negative phase shift results in a move to the left.
• This horizontal movement is crucial, especially when comparing or synchronizing waves.

Understanding phase shifts helps in sketching the sine wave correctly, showing how the wave is positioned in relation to the y-axis. It's like setting the starting point of the wave.