Question

Graph each sine wave. Find the amplitude, period, and phase shift.$$y=2 \sin \left(x-35^{\circ}\right)$$

Short answer

The amplitude is 2, the period is 360 degrees, and the phase shift is \(+35^\circ\) to the right.

Step-by-step solution

Identify the Amplitude

The amplitude of a sine wave is the coefficient in front of the sine function. In this case, the amplitude is the absolute value of 2, so the amplitude is 2. The amplitude indicates the maximum value that the sine wave reaches, both above and below the x-axis.

Identify the Period

The period of a sine wave is the distance required for the function to complete one full cycle. For the basic sine function, the period is \(2\pi\) radians. There is no coefficient in front of the \((x - 35^\circ)\) that modifies the period, so the period remains \(2\pi\) radians or 360 degrees.

Identify the Phase Shift

The phase shift is determined by the horizontal shift inside the sine function. It is in the form of \(x - C\) where \(-C\) is the phase shift. Here, it's \(x - 35^\circ\), thus the phase shift is \(+35^\circ\) to the right.

Graphing the Sine Wave

To graph the sine wave, start by plotting the phase shift on the x-axis. With an amplitude of 2, mark points at 2 and -2 on the y-axis to guide the peak and trough of the wave. Since the period is 360 degrees, one complete cycle will be from \(+35^\circ\) to \(395^\circ\) on the x-axis. Draw the sine curve starting at 35 degrees, peaking at \((35^\circ + 90^\circ) = 125^\circ\), crossing the x-axis at \((35^\circ + 180^\circ) = 215^\circ\), reaching the trough at \((35^\circ + 270^\circ) = 305^\circ\), and finishing the cycle at 395 degrees.

Key concepts

Amplitude of Sine Wave

When exploring the concept of sine waves, the amplitude is a fundamental aspect to understand. It represents the peak value of the wave; that is, how far the wave oscillates from its central axis or equilibrium position. In the equation \(y = 2\sin(x - 35^\circ)\), the number in front of the sine function, which is 2, indicates the amplitude.

To visualize this, picture a calm sea with waves flowing gently—the height of these waves from the still water level to the crest is what the amplitude would represent in this analogy. The higher the amplitude, the more intense the wave. Conversely, a smaller amplitude corresponds to gentler waves. In practical contexts, like audio engineering, amplitude can determine the volume of a sound, while in light waves, it could influence brightness.

• The amplitude of our given sine wave is 2, which means the wave's crests and troughs are 2 units above and below the x-axis respectively.
• This information is crucial when graphing, as it dictates the maximum and minimum points you should plot for the sine wave.

By understanding amplitude, students can better depict the impact of this variable on the overall wave and its graph.

Period of Sine Wave

Diving deeper into wave characteristics, the period of a sine wave becomes an essential concept. The period represents the length of one full cycle of the wave—think of it like the time it takes for a swing to return to its starting point after being pushed. In sine wave terms, it's the horizontal distance needed for the wave to repeat its pattern.

A standard sine wave, which is written as \(y = \sin x\), completes its cycle every \(2\pi\) radians or 360 degrees. In our function \(y = 2\sin(x - 35^\circ)\), there are no additional coefficients stretching or compressing the wave horizontally, so the period remains unchanged at \(2\pi\) radians.

• We identify the period as a critical parameter for graphing because it tells us the width of each wave cycle on the x-axis.
• A knowledge of the period aids in predicting where the wave will begin repeating its shape, marking out a fundamental pattern unit.

Understanding the period is important for students interpreting functions as it influences the 'spacing' of the cycles and their frequency.

Phase Shift of Sine Wave

Lastly, the concept of phase shift must not be overlooked. Think of phase shift as the placement of a wave along the horizontal axis—if you could slide the wave left or right, the amount you move it by would be the phase shift. It shows how the wave's position is being adjusted from its standard starting point.

The phase shift of a sine wave is determined by the horizontal translation inside the parentheses of the sine function. It’s the value that 'shifts' the standard \(y = \sin x\) wave to the left or the right on a graph. For our example \(y = 2\sin(x - 35^\circ)\), the phase shift is 35 degrees to the right because the function has \(x - 35^\circ\), indicating a positive shift.

• This adjustment is crucial for accurately graphing the wave's starting point.
• Grasping the concept of phase shift is vital for aligning waves in applications such as signal processing or understanding the superposition of waves.

Recognizing the phase shift helps students manipulate and predict the position of sine waves, a skill that is crucial for numerous scientific and engineering tasks.