Period and Phase Shifts
The period of a trigonometric function is defined to be when the graph repeats itself.
In most cases, the period or cycle is shown from a starting point such as where x = 0 or perhaps from a max or min value, but that's not required.
The standard period of a purely sinusoidal function (Sine and Cosine) is pretty easy to identify as being 2 pi.
The period of Tangent and Cotangent is also pretty easy to identify, but whether or not these are purely sinusoidal functions is not set in stone. Some say Yes because they repeat. Some say No because the functions are not continuous because of the vertical asymptotes.
That being said, Tangent and Cotangent do repeat, so their period can be identified.
Identifying the Period visually is nothing more than determining how long it takes for a graph to repeat itself.
Identifying the period of a function graphically is pretty easy.
Calculating the period of a function graphically is so easy, we rarely see those types of problems. Instead, we end up having to solve them algebraically.
Find the Amplitude and Period of each of the following functions:
Calculating the Period algebraically requires a few steps.
Identify the Periodicity factor in your function definition. The Periodicity value is often defined as the 'B' value.
It is the multiplier inside the parenthesis of the Trig function and right in front of the angle.
It is the multiplier inside the parenthesis of the Trig function and right in front of the angle.
Examples:
Once you have identified things in your function, such as Amplitude, Period, and Vertical Shift and you have to graph it.....
It's important to understand that the Amplitude, Period, and Vertical Shift does not fundamentally change the basic function.
If you are working with a Sine wave, it will still start at the origin (0,0).
If you are working with a Cosine wave, it will start at (0,1).
If you are working with Tangent, it will start at -pi/2.
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Consider the following diagram. It's not too difficult to notice that Sin and Cosine are pretty closely related to each other.
In fact, if we overlay the two functions, we get the following:
The functions are identical, they're just shifted sideways a little. This sideways shifting is known as 'Phase Shift'.
Phase Shift is the amount that a curve is shifted left or right from its normal position.