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(**b**x +**c**) +**d** - y =
**a**⋅cos(**b**x +**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 = *x*_{final} – *x*_{initial}. 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:

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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.