Question

Graph each sine wave. Find the amplitude, period, and phase shift.$$y=3 \sin 2 x$$

Short answer

The amplitude is 3, the period is \(\pi\), and there is no phase shift. Sketch a sine wave with these properties starting at the origin, oscillating between 3 and -3 on the y-axis and repeating every \(\pi\) units on the x-axis.

Step-by-step solution

Identify Amplitude

The amplitude of a sine wave is the coefficient of the sine function. In the equation \(y=3\sin(2x)\), the coefficient is 3, which indicates that the amplitude is 3. This means the wave height from the center line to the peak or to the trough is 3 units.

Determine the Period

The period of a sine function is given by \(\frac{2\pi}{B}\) where \(B\) is the coefficient of \(x\) inside the sine function. For the given equation \(y=3\sin(2x)\), \(B=2\), so the period is \(\frac{2\pi}{2} = \pi\). This means that the wave completes one full cycle over an interval of length \(\pi\) on the x-axis.

Find the Phase Shift

The phase shift of a sine function is determined by any horizontal shift present in the function. For \(y=3\sin(2x)\), there is no horizontal shift, hence the phase shift is 0.

Sketch the Sine Wave

Using the amplitude and period, start to sketch the sine wave. The wave oscillates between 3 and -3 on the y-axis, and completes a full cycle every \(\pi\) units on the x-axis. Since there is no phase shift, the wave starts at the origin. Mark the key points: a peak at \(\left(\frac{\pi}{4}, 3\right)\), crossing the x-axis at \(\left(\frac{\pi}{2}, 0\right)\), a trough at \(\left(\frac{3\pi}{4}, -3\right)\), and crossing the x-axis again at \(\left(\pi, 0\right)\). Continue plotting the wave to the right and left as needed.

Key concepts

Amplitude of a Sine Wave

When graphing sine waves, understanding the amplitude is crucial. It represents the height of the wave's peak (crest) or the depth of its trough (valley) from the wave's center line. In a mathematical equation like
\(y=3\sin(2x)\), it's the absolute value of the coefficient in front of the sine function, which in this case is 3. This tells us that the wave oscillates 3 units above and below the horizontal axis, which serves as our center line for the wave.

If you were to increase the amplitude, imagine the wave's peaks and valleys stretching further away from the center line, making them more prominent. On the contrary, reducing the amplitude makes the wave flatter, as it's not reaching as high above or dipping as low beneath this center line. The concept of amplitude is a measure of the energy in the wave, with higher amplitudes signaling more energetic waves.

Period of a Sine Wave

The period is another fundamental feature of sine waves and tells you how wide one cycle of the wave is. It's the distance on the x-axis between two consecutive points where the wave starts to repeat itself, like from peak to peak or trough to trough.
For the equation \(y=3\sin(2x)\), we look at the coefficient of \(x\) inside the sine function to determine the period. Since our coefficient, \(B\), is 2, we calculate the period by \(\frac{2\pi}{B} = \frac{2\pi}{2} = \pi\). This means that the wave completes its circuit—a return to the starting position—every \(\pi\) units along the x-axis.

It's fascinating how the period gives us a rhythm or tempo for the wave, like beats in a music track, telling us the length of one full note or wave cycle before it repeats the next.

Phase Shift of a Sine Wave

A less immediately apparent but equally important characteristic of sine waves is the phase shift. It measures the horizontal shift of the wave along the x-axis. Essentially, it tells us if the typical sine wave pattern has been slid to the left or right from its usual starting point at the origin (0,0).
In our specific example of \(y=3\sin(2x)\), scrutinizing the interior of the sine function reveals no added or subtracted constants, meaning there is no phase shift, and our wave begins at the origin as expected. However, if we were to see an equation like \(y=3\sin(2x - \pi/2)\), there would be a phase shift of \(\pi/2\) units to the right.

Being able to identify and calculate the phase shift is like knowing when to start dancing to the wave's rhythm, ensuring you're in sync from the very first beat.