1. Graphs of Trigonometric Functions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Properties of Sine and Cosine Functions 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.

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Sine Function 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

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Cosine Function 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

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 y x Example: y = 3 cos x Example: 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 30-303 y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Amplitude 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

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 y x Period of a Function 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 b > 1, the graph of the function is shrunken horizontally. If 0 < b < 1, the graph of the function is stretched horizontally. y x period: 2 period: 4

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Graph y = f(-x) Example : Sketch the graph of y = - sin (x). The graph of y = - sin (x) is the graph of y = sin x reflected in the x-axis. y x y = sin x y = sin (–x)

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 y x 0 20 –2 0y = –2 sin 3x 0 x Example: y = 2 sin(-3x) Example: Sketch the graph of y = - 2 sin (3x). The function in the form y = a sin bx with b > 0 amplitude: |a| = |–2| = 2 Calculate the five key points. (0, 0) (, 0) (, 2) (, -2) (, 0) y = - 2 sin (–3x) period: 2 2 3 =

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 a = the height between the max/min and the midline. If negative, the graph reflects over its midline b = affects the period. In fact, c = affects the phase shift (i.e. horizontal). Shifts graph in opposite direction of the sign in the parentheses. d = affects the vertical shift. It moves the midline up (if positive) and down (if negative).

11. Sketching A Sinusoid Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

12. Examples Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12