Graphs of Sine and Cosine

Label your graph paper... 2 1 90º -270º 270º 360º -90º -1 180º -2

In radians... 2 1  2 -1 -2

Characteristics of the Sine Function 1. The domain is the set of all real numbers. 2. The range consists of all real numbers from -1 to 1, inclusive. 3. The sine function is an odd function (symmetric with respect to the origin). 4. 5.

Characteristics of the Sine Function 6.

Graph y = sin  sin  0° 0.707 45° 90° 1 135° 0.707 180° 225° 2 0° 0.707 45° 1 90° 1 -270º -90º 90º 180º 270º 360º 135° 0.707 -1 180° -2 225° -0.707 -1 270° -0.707 315° 360°

y = sin x 2 1 -90º 90º -270º 270º -1 -2 Maximum Maximum intercept Minimum Minimum -2

y = sin x Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1 -90º 90º -270º 270º -1 -2 Period: 360°

In other words, how high does it go from its axis? y = sin x Amplitude: half the distance between the maximum and minimum 2 Amplitude = 1 1 -90º 90º -270º 270º -1 -2 In other words, how high does it go from its axis?

Characteristics of the Cosine Function 1. The domain is the set of all real numbers. 2. The range consists of all real numbers from -1 to 1, inclusive. 3. The cosine function is an even function (symmetric with respect to the y-axis). 4. 5.

Characteristics of the Cosine Function 6.

Graph y = cos cos   1 0.707 -0.707 -1 -0.707 0.707 1 2 1 -1 -2 1 2 0.707 1 -0.707  2 -1  -1 -0.707 -2 0.707 2 1

y = cos x -2 -  2 2 1 -1 -2 Maximum Maximum intercept intercept Minimum Minimum -2

y = cos x -2 -  2 Period: 2 Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1 -2 -  2 -1 -2 Period: 2

How high does it go from its axis? y = cos x How high does it go from its axis? 2 Amplitude = 1 1 -2 -  2 -1 -2

Try it on your calculator! y = sin x y = cos x Try it on your calculator! 2 1 -1 -2

y= sin and y = cos are the mother functions. Changing the equations changes the appearance of the graphs We are going to talk about the AMPLITUDE, TRANSLATIONS, and PERIOD of relative equations

Mother Function relative function change? y1 = sin x reflection over x-axis y2 = - sin x y1 = sin x y2 = 4 sin x amplitude = 4 amplitude = y2 = sin x y1 = sin x generalization? y = a sin x amplitude = a

is the horizontal translation Mother Function relative function change? y1 = sin x y2 = sin (x - 45) horizontal translation, 45 degrees to the right. horizontal translation, 60 degrees to the left. y1 = sin x y2 = sin (x + 60) horizontal translation, 30 degrees to the left. y2 = sin (2x + 60) y1 = sin x horizontal translation, 90 degrees to the right. y1 = sin x y2 = sin (3x - 270) generalization? y = sin (bx - c) to the right is the horizontal translation y = sin (bx – (- c)) to the left

‘d’ is the vertical translation Mother Function relative function change? y1 = cos x y2 = 2 + cos x vertical translation, 2 units up. vertical translation, 3 units down. y1 = cos x y2 = -3 + cos x generalization? y = d + cos x ‘d’ is the vertical translation when d is positive, the graph moves up. when d is negative, the graph moves down.

Mother Function relative function change? y1 = sin x y2 = sin 2x Period = 180 or Period = 720 or y1 = sin x y2 = sin x generalization? Period = y = sin bx or

Summary: y = d + a sin (bx - c) y = d + a cos (bx - c) a is the amplitude period = or is the horizontal translation d is the vertical translation

Analyze the graph of amplitude = period = horizontal translation: vertical translation: none

Analyze the graph of 3 (to the left) amplitude = period = horizontal translation: vertical translation: none

Analyze the graph of 3 amplitude = period = horizontal translation: none vertical translation: Up 2

Graph and Analyze y = -2 + 3 cos (2x - 90°) 3 x y high 1 amplitude = 45° 3 amplitude = 90° -2 mid period = = 180° 135° low -5 horizontal translation: 180° -2 mid (to the right) 225° 1 high vertical translation: down 2 1) horiz. tells you where to start 3) divide period by 4 to find increments 180 4 = 45 2) add the period to find out where to finish table goes in increments of 45 4) plot points and graph 45 + 180 = 225

You must know how to analyze the equation before you can graph it. The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table.

Ex #6c Graph y = 1 + 3 sin (2 + ) =  3 up 1 x y mid 1 amplitude = high =  4 period = 1 mid horizontal translation: -2 low up 1 vertical translation: mid 1 1) horiz. tells you where to start On Calculator, go to table setup and change independent to ask. 3) divide period by 4 to find increments table goes in increments of 2) add the period to find out where to finish 4) plot points and graph