y = sin(x) and y = cos(x) are periodic functions because all possible y values repeat in the same sequence over a given set of x values. The “length” of this interval of x values is called the period. To refresh your memories on these periodic functions, review this lesson on their graphs. The general form of a sine or cosine function is given in two different ways: Method IMethod IIthe amplitude is Athe amplitude is Athe vertical shift is Dthe vertical shift is Dthe period is the period is B = where f is the frequency B = where f is the frequency the phase shift is Cthe phase shift is These functions are often shifted vertically or horizontally .
Carefully inspecting the equation of f(x) tells us that A = 1 B = 2 C = D = 2 We can now calculate the following: period = phase shift = . The graph of this function is shown below with a WINDOW of X: and Y: (-2, 4, 1).
The dotted line is Y = D = 2 and serves as the horizontal axis. The point plotted has coordinates and serves as a “starting point” for a sine graph shifted units to the right.
Carefully inspecting the equation f(x) tells us that A = 3C = D = -2 We can now calculate the following: period = frequency = the reciprocal of the period = 2 This can also be determined with the formula B = . The phase shift is . The graph of this function is shown below with a WINDOW of X: and Y: (-6, 2, 1). The dotted line is the horizontal axis is Y = -2The point plotted is and serves as the "starting point" for a cosine graph shifted unit to the right.
This can also be determined with the formula B = .