Section 4.5 Graphs of Sine and Cosine

Overview In this section we first graph y = sin x and y = cos x. Then we graph transformations of sin x and cos x by applying changes in amplitude, period, and phase shift.

Some helpful information The period of both sine and cosine is 2π. This means that, after 2π the values of sine and cosine begin to repeat themselves. The domain of both sine and cosine is “all real numbers”. The range of both sine and cosine is [-1,1].

The graphs of sine and cosine We will use our completed table to get the necessary ordered pairs. The x-coordinates are the angles, measured in radians. The y-coordinates are the values of sine or cosine, depending on the function being graphed. Note particularly what the graphs do at 0, π/2, π, 3π/2, and 2π. These five points are key because they occur at the beginning, ¼ of the way, ½ of the way, ¾ of the way, and back at the beginning.

Sinusoidal Graphs Graphs of sine and cosine are generally referred to as sinusoidal graphs. The general form of the equations are

The Amplitude Amplitude refers to how high and how low the graphs goes. The amplitude of a sinusoidal graph is |A|.

Examples Find the amplitude of each of the following:

The Period Period refers to the length of the graph before the values start to repeat themselves. The period of a sinusoidal graph is given by

Examples Find the period of each of the following:

Phase Shift Phase shift refers to a horizontal translation of all the points on the graph. The value of the phase shift is x = Look at the sign in your parentheses: if the sign is (+), shift the graph to the left. If the sign is (-), shift the graph to the right.

Five Steps 1.Identify the amplitude, the period, and the phase shift. 2.Divide the period into four equal parts. Use ¼, ½, and ¾ to get the values of x. 3.If the phase shift is to the left, subtract from each of the x-values. If the phase shift is to the right, add to each of the x-values. 4.Note the behavior of the graph at each x-value (intercept, maximum, or minimum). Be sure to take amplitude into account. Plot the points.

Examples Find the phase shift of each of the following: