Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\sin (2 x-\pi)+1 $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \( \pi \), Phase Shift: \( \frac{\pi}{2} \) right.

Step 1: Understand the Standard Form

The given equation is \( y = \sin(2x - \pi) + 1 \). The standard form of the sine function is \( y = A \sin(Bx - C) + D \), where \( A \) is the amplitude, \( \frac{2\pi}{B} \) is the period, \( \frac{C}{B} \) is the phase shift, and \( D \) is the vertical shift.

Step 2: Identify the Amplitude

The amplitude is the coefficient of the sine function in the standard form. Here, the amplitude \( A \) is 1 because the coefficient in front of \( \sin(2x - \pi) \) is 1.

Step 3: Calculate the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{B} \). Here, \( B = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Step 4: Determine the Phase Shift

The phase shift is calculated by \( \frac{C}{B} \), where \( C \) is the constant subtracted from \( 2x \). Thus, \( C = \pi \). So, the phase shift is \( \frac{\pi}{2} \), and since the equation is \( \sin(2x - \pi) \), the shift is to the right.

Step 5: Identify the Vertical Shift

The vertical shift is given by the constant \( D \) added to the sine function. Here, \( D = 1 \), so the graph is shifted 1 unit upwards.

Step 6: Sketch the Graph

First, plot the basic sine wave with a period of \( \pi \), an amplitude of 1, and a rightward phase shift of \( \frac{\pi}{2} \). Then, adjust the graph to reflect the vertical shift of 1 unit up. Include key points at \( (\frac{\pi}{2}, 1), (\frac{3\pi}{4}, 2), (\pi, 1), (\frac{5\pi}{4}, 0), (\frac{3\pi}{2}, 1) \). The graph should now show these movements.

Key Concepts

Amplitude

Amplitude in trigonometric functions refers to the height of the wave from the centerline to its peak. For a sine wave, this represents how high and low the wave goes from its average position. Amplitude is always a positive value and is symbolized as \( A \) in the function.

In the given equation \( y = \sin(2x - \pi) + 1 \), the coefficient in front of the sine function determines the amplitude.
Since there isn't an explicit number before the sine, it means the coefficient is 1 by default. Thus, the amplitude here is simply 1.

Understanding amplitude is crucial, as it affects the vertical stretch or compression of the sine wave.

• An amplitude greater than 1 means the wave stretches vertically.
• An amplitude less than 1 compresses the wave vertically.
• In this equation, with an amplitude of 1, the wave stays in its normal pattern, moving from -1 to 1 on the y-axis, but because of the vertical shift, it moves between 0 and 2 instead.

Period

The period of a trigonometric function measures how long it takes for the function to complete one full cycle. For a standard sine wave, the period is \( 2\pi \).

A change in the function’s period changes the wave’s width horizontally. The period is given by the formula \( \frac{2\pi}{B} \), where \( B \) is the frequency of the function.

In the equation \( y = \sin(2x - \pi) + 1 \), the value of \( B \) is 2. This implies that the period is \( \frac{2\pi}{2} = \pi \).

Understanding how to calculate and interpret the period is key for sketching the graph accurately:

• A smaller period than usual means the wave cycles faster or has more waves within the same length of the x-axis.
• A longer period would stretch the graph, reducing the number of cycles.
• Here, the wave finishes a complete cycle by the time \( x \) reaches \( \pi \), which is shorter than a typical sine wave.

Phase Shift

Phase shift describes the horizontal movement of a wave, shifting it left or right without altering its shape. This often represents moving a wave to match certain conditions. The phase shift is calculated using \( \frac{C}{B} \).

In the equation \( y = \sin(2x - \pi) + 1 \), \( C \) is represented by the constant \( \pi \), and with \( B = 2 \), the phase shift calculation becomes \( \frac{\pi}{2} \).
This shift to the right is because \( C \) is subtracted, moving the starting point of the sine wave over to match \( x = \frac{\pi}{2} \).

Correctly identifying the phase shift is essential for aligning the function's graph with the desired position on the x-axis:

• A positive phase shift (as in this case) means the function moves to the right.
• A negative phase shift would move it left.
• The phase shift does not alter the amplitude or period; it just repositions the wave.

Understanding phase shifts helps in accurately placing key points on the graph, reflecting where the wave peaks or zeros occur relative to the y-axis.