The final section of this chapter involves making an equation from a waveform. In other words, you’ll be shown a wave, and you’ll be expected to identify its amplitude, period, and phase-shift, then use this information to generate an equation using one of the templates shown below. Since a sine wave can technically be written in terms of cosine, and vice-versa, the question will specify whether to write it in terms of sine or cosine.
- y = a⋅sin(bx + c) + d
- y = a⋅cos(bx + c) + d
The easiest of the three properties stated above to find is the amplitude ‘a’; it is the distance from the center of the wave to the y-coordinate of the maximum or minimum point. To find the period, you must locate the start and end points of a cycle. You then calculate the difference between their x-coordinates, where the period = xfinal – xinitial. The period can be used to find the factor ‘b’ using the formula shown below. Lastly, the phase-shift can be found by visual inspection. Simply locate the distance between where the wave begins to the vertical axis, where x = 0. You can use the phase-shift that you find, along with the value for ‘b’ found in the step prior, to find ‘c’ using the formula below. Keep in mind that reflections are difficult to interpret if you’re given multiple cycles of a waveform. If you’re given a single cycle and you’re told there’s a reflection, ‘a’ can be made negative.
The first two videos caters to creating sinusoidal functions with cosine.
What’s interesting about the answer found – and generally for all equations you find for these types of questions – is that more than one equation can represent the waveform. That is, you can have variations of ‘c’ by adding or subtracting the period with the phase-shift. Take, for example, the equation that was found above: y = 5·cos(x – 3π/5). The period was found to be 2π, and the phase-shift 3π/5.
Adding (or subtracting) 2π to the phase-shift gives a new phase-shift:
Solving for ‘c’ once more using phase-shift = –c / b, gives you a new variant of your original equation:
- y = 5⋅cos(x – 13π/5)
- y = 5⋅cos(x + 7π/5)
The reason why this works is because choosing the phase-shift is somewhat of a subjective decision, especially if you’re given multiple cycles of the wave.
The next two videos show a second example of a cosine equation being created, while the final video shows a sine equation.