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6.4 Graphs of Sine and Cosine. 90º 270º 180º 1 2 -2 360º -270º -90º Label your graph paper...

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Presentation on theme: "6.4 Graphs of Sine and Cosine. 90º 270º 180º 1 2 -2 360º -270º -90º Label your graph paper..."— Presentation transcript:

1 6.4 Graphs of Sine and Cosine

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

3  22 In radians...

4 Graph y = sin  90º-90º 270º-270º -2  sin  0°0° 45° 90° 135° 180° 225° 270° 315° 360°   º360º 2 1

5 y = sin x 90º-90º 270º-270º Maximum Minimum intercept

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

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

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9 Graph y = cos   cos      22  22

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

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

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

13 y = cos xy = sin x Try it on your calculator!

14 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

15 Mother Functionrelative functionchange? y 1 = sin x y 2 = - sin x reflection over x-axis y 1 = sin x y 2 = 4 sin x y 2 = sin x amplitude = 4 amplitude = generalization? y = a sin x amplitude = a

16 Mother Functionrelative functionchange? y 1 = sin x y 2 = sin (x - 45) y 2 = sin (x + 60) horizontal translation, 45 degrees to the right. horizontal translation, 60 degrees to the left. y 1 = sin x generalization? y = sin (bx - c) y = sin (bx – (- c)) is the horizontal translation to the right to the left y 2 = sin (2x + 60) y 1 = sin x horizontal translation, 30 degrees to the left. y 2 = sin (3x - 270)y 1 = sin x horizontal translation, 90 degrees to the right.

17 Mother Functionrelative functionchange? y 1 = cos x y 2 = 2 + cos x vertical translation, 2 units up. y 1 = cos xy 2 = -3 + cos x vertical translation, 3 units down. 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.

18 Mother Functionrelative functionchange? y 1 = sin x y 2 = sin 2x y 2 = sin x Period = 180 or Period = 720 or generalization? y = sin bx Period = or

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

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

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

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

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

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

26 y = sin (2  +  ) amplitude = period = vertical translation: horizontal translation: xy ) horiz. tells you where to start 2) add the period to find out where to finish 3) divide period by 4 to find increments 4) plot points and graph Ex #6cGraph =  3 up 1 table goes in increments of mid high low

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