Thus, Figure can also represent the graph of y = −3 sin 2 x.
The graph of the function y = − f( x) is found by reflecting the graph of the function y = f( x) about the x‐axis. In other words,Įxamples of several phase shifts of the sine function.Įxample 3: What is the amplitude, period, phase shift, maximum, and minimum values ofĮxample 4: Sketch the graph of y = cosπ x.īecause cos x has a period of 2π, cos π x has a period of 2 (Figure 9).Įxample 5: Sketch the graph of y = 3 cos (2x + π/2).īecause cos x has a period of 2π, cos 2x has a period of π (Figure 10). The cosine function looks exactly like the sine function, except that it is shifted π/2 units to the left (Figure 8). The sine function is odd, and the cosine function is even. It does not matter whether the shift is to the left (if D is positive) or to the right (if D is negative). (This also holds for the cosine function.) The phase shift is | D |. The additional term D in the function y = sin ( x + D) allows for a phase shift (moving the graph to the left or right) in the graph of the sine functions. Figure 7 illustrates additional examples.Įxamples of several frequencies of the a) sine function and b) cosine function. Thus, the function y = sin 5 x has a period of 2π/5. (This also holds for the cosine function.) The period of the function y = sin Cx is 2π/|C|. The additional factor C in the function y = sin Cx allows for period variation (length of cycle) of the sine function. The minimum value is 4 − 3 = 1 (Figure 6 ). What are the maximum and minimum values of the function? The minimum value is 1 −2 = −1 (Figure 5 ).Įxample 2: Graph the function y = 4 + 3 sin x. This minimum occurs whenever sin x = −1 or cos x = −1.Įxample 1: Graph the function y = 1 + 2 sin x. The minimum value of the function is m = A ‐ |B|. This maximum value occurs whenever sin x = 1 or cos x = 1. The maximum value of the function is M = A + |B|. These two functions have minimum and maximum values as defined by the following formulas. This also holds for the cosine function (Figure 4 ).Įxamples of several amplitudes of the sine function.Ĭombining these figures yields the functions y = A + B sin x and also y = A + B cos x. The amplitude, | B |, is the maximum deviation from the x‐axis-that is, one half the difference between the maximum and minimum values of the graph.
The additional factor B in the function y = B sin x allows for amplitude variation of the sine function. This also holds for the cosine function (Figure 3 ).Įxamples of several vertical shifts of the sine function. The additional term A in the function y = A + sin x allows for a vertical shift in the graph of the sine functions. Several additional terms and factors can be added to the sine and cosine functions, which modify their shapes. Multiple periods of the a) sine function and b) cosine function. The sine function and the cosine function have periods of 2π therefore, the patterns illustrated in Figure are repeated to the left and right continuously (Figure 2 ). One period of the a) sine function and b) cosine function. Next, plot these values and obtain the basic graphs of the sine and cosine function (Figure 1 ). To see how the sine and cosine functions are graphed, use a calculator, a computer, or a set of trigonometry tables to determine the values of the sine and cosine functions for a number of different degree (or radian) measures (see Table 1).