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The amplitude of a sine function represents the maximum distance the wave reaches from its central position (midline). For a sine function written in the form \( y = A \sin(Bx + C) + D \), the amplitude is \( |A| \). In this exercise, the function is \( y = \frac{1}{3} \sin(x) - 4 \), so \( A = \frac{1}{3} \). Therefore, the amplitude is: \( \frac{1}{3} \) This means the sine wave's peaks are \( \frac{1}{3} \) units above the midline and the troughs are \( \frac{1}{3} \) units below the midline.

The period of a sine function indicates the length of one complete cycle of the wave. For the general form of a sine function: \( y = A \sin(Bx + C) + D \)), the period is computed using the formula: \( \frac{2\pi}{B} \) In our exercise, the function \( y = \frac{1}{3} \sin(x) - 4 \) has \( B = 1 \), therefore the period is: \( \frac{2\pi}{1} = 2\pi \) This tells us that it takes \( 2\pi \) units along the x-axis for the sine wave to complete one full cycle.

Phase shift is the horizontal displacement of the sine wave along the x-axis. It is given by the formula: \( \frac{-C}{B} \) Using the function: \( y = \frac{1}{3} \sin(x) - 4 \), we identify \( C = 0 \) and \( B = 1 \), so the phase shift is: \( \frac{-0}{1} = 0 \) This means there is no phase shift. The wave starts at \( x = 0 \), so no horizontal adjustment is needed.

The vertical shift moves the entire graph up or down along the y-axis. In the sine function \( y = A \sin(Bx + C) + D \), the vertical shift is represented by \( D \). For example: \( y = \frac{1}{3} \sin(x) - 4 \), we have \( D = -4 \). Therefore, the graph is shifted down by 4 units. The midline of the sine wave is now situated at \( y = -4 \) instead of the x-axis.

Graphing the sine function involves plotting key points based on the amplitude, period, phase shift, and vertical shift: 1. Draw a horizontal line at \( y = -4 \) to represent the midline. 2. Using a range from \( 0 \) to \( 2\pi \), identify key points: - Start at \( x = 0 \), \( y = -4 \) - Maximum at \( x = \frac{\pi}{2} \), \( y = -3.66 \) from \( \frac{1}{3} \) above the midline - Return to baseline at \( x = \pi \), \( y = -4 \) - Minimum at \( x = \frac{3\pi}{2} \), \( y = -4.33 \), \( 4.33 \) from below the midline - End at \( x = 2\pi \), \( y = -4 \) 3. Connect the points with a smooth, sinusoidal curve. Verify your sketch using a graphing calculator to check accuracy. Happy graphing!