Chp. 4.5 Graphs of Sine and Cosine Functions p. 323

In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are positive constants and x is in radian measure. The graphs of all sine and cosine functions are related to the graphs of y = sin x and y = cos x which are shown below. y = sin x y = cos x

x Sin x Cos x Fill in the chart. These will be key points on the graphs of y = sin x and y = cos x.

6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.

Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1001cos x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x

Before sketching a graph, you need to know: Amplitude – Constant that gives vertical stretch or shrink. Period – Interval – Divide period by 4 Critical points – You need 5.(max., min., intercepts.)

Amplitudes and Periods The graph of y = A sin Bx has amplitude = | A| period = The graph of y = A sin Bx has amplitude = | A| period = To get your critical points (max, min, and intercepts) just take your period and divide by 4. Example: Interval

The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2sin x y = sin x Notice that since all these graphs have B=1, so the period doesn’t change.

y x period: 2 period: The period of a function is the x interval needed for the function to complete one cycle. For b  0, the period of y = a sin bx is. For b  0, the period of y = a cos bx is also. If 0 < b < 1, the graph of the function is stretched horizontally. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4

y x Example 1: Sketch the graph of y = 3 cos x on the interval [– , 4  ]. Partition the interval [0, 2  ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

Determine the amplitude of y = 1/2 sin x. Then graph y = sin x and y = 1/2 sin x for 0 < x < 2 . Example 2

x y 2˝ ˝ 1 y = sin x y = 1/2sinx

Example 3

For the equations y = a sin(bx-c)+d and y = a cos(bx-c)+d a represents the amplitude. This constant acts as a scaling factor – a vertical stretch or shrink of the original function. Amplitude = The period is the sin/cos curve making one complete cycle. Period = c makes a horizontal shift. d makes a vertical shift. The left and right endpoints of a one-cycle interval can be determined by solving the equations bx-c=0 and bx-c=

Example 4

Example 6