Math-3. Lesson 6-8. Graphs of the sine and cosine functions; and Periodic Behavior

Transcription

1 Math-3 Lesson 6-8 Graphs o the sine and cosine unctions; and Periodic Behavior

2 What is a unction? () Function: a rule that matches each input to eactly one output. What is the domain o a unction? Domain: the set o all allowable input values o a unction (the values that have corresponding y values). What is the range o a unction? Range: the set o all output values o a unction (the y values).

3 The sine unction (as opposed to the sine ratio) r = y (, ( ) sin Sine o the angle is the y-value o the point on the -y plane where the terminal side o the angle intersects the unit circle. y) sin 0 opp hyp y y 70

4 The sine unction (as opposed to the sine ratio) r = 70 y (, ( ) sin What is the input to the unction? angle What is the output o the unction? y) What is the maimum Ө-value? What is the minimum Ө-value? 0 y-value o the point - What is the maimum y-value? What is the minimum y-value? -

5 Graph o the Sine Function ( ) sin Input value: the angle Output value: the y-value o the point on the circle corresponding to the angle. radians

6 + y The Sine Function The input value is the angle. The input value is plotted on the -ais. The output value is the ratio (opp/hyp) (Input) θ ( ) sin (rule) sin(θ) 90 Sin (90) (output) (θ) 0 Sin (0) 0 30 Sin (30) Sin (45) Sin (60) Sin (0) Sin (35) Sin (50) Sin (80) 0

7 Graph o the Sine Function Think o a dot traveling around the circle to the right..5 radians radians - -.5

8 Graph o the Sine Function:.5 radians radians Think o sin θ as the distance above (or below) the -ais determined by θ (the angle).

9 Sinusoid ( ) sin Domain =? Range =? y -intercepts? n * n integers

10 The Cosine unction (as opposed to the cosine ratio) r = y (, y) ( ) cos cos adj hyp cosine o the angle is the -value o the point where the terminal side o the angle intersects the unit circle.

11 The cosine unction (as opposed to the cosine ratio) r = 70 y (, y) 0 ( ) cos What is the input to the unction? angle What is the output o the unction? y-value o the point What is the maimum Ө-value? What is the minimum Ө-value? - What is the maimum y-value? What is the minimum y-value? -

12 Graph o the Cosine Function ( ) cos Input value: the angle Output value: the -value o the point where the terminal side o the angle intersects the unit circle. radians

13 Graph o the Cosine Function ( ) cos cos().5 radians Think o a dot traveling around the circle to the right.

14 Graph o the Cosine Function ( ) cos cos().5 radians Think o cos θ as the distance to the right (or let) o the y-ais as determined by θ (the angle).

15 Sinusoid ( ) cos Domain =? Range =? y -intercepts? cos( 90) * n n integers sin( )

16 Sinusoid ( ) cos Domain =? Range =? y -intercepts? n n integers cos( ) sin( )

17 When the angle is between 0 and 90 (quadrant ) sin sin,cos, tan cos tan I What is the sign (+/-) o: 0 70

18 Your turn: Match the graph with its unction: () = sin θ g() = cos θ a. b.

19 The Transormation Equation y Vertical relection stretch shit ( )* a* ( h) k shit ( ) Relection across -ais ( ) a Vertical stretch ( ) h Horizontal shit ( ) k Vertical shit

20 Describe the transormations these unction make on their parent unctions. ( ) g( ) 4 5 Vertically stretched by a actor o, shited right 4 and up 5 h( ) k( ) 0.5( ) 6 Vertically stretched by a actor o 0.5, shited let and down 6

21 Compare: ( ) asin Amplitude: For the sine and cosine unctions, we call the coeicient a the amplitude o the unction (which in general reers to the vertical stretch actor). -

22 Compare: ( ) sin g( ) sin

23 g( ) sin ( ) sin shit Centerline o the Oscillation: corresponds to the up/down translation.

24 ( ) g( ) 4 ( ) g( ) 4 Vertical Stretch: multiplying the original unction by 4, vertically stretches it by a actor o 4. Horizontal Stretch: replacing with a number multiplied by, h( ) () 4 For the square unction a vertical Stretch is the same as a horizontal shrink (so you really can t tell the dierence between them).

25 g( ) sin Predict what you think happens Horizontal stretch or shrink? ( ) sin horizontal shrink Stretched by a actor o ½ OR Shrunk by a actor o.

26 Vocabulary Period: the horizontal distance needed to complete one ull cycle o the oscillation. g( ) sin Period

27 Period: the horizontal distance needed to complete one ull cycle o the oscillation. g( ) sin ( ) sin horizontal stretch actor Period *

28 Compare: g( ) ( ) sin What is the period o g()? asin b ( ) sin 3 /b horizontal stretch shrink actor =? 3 What is the period o ()? b * b 3

29 Your turn: ( ) asin b ( ) 4cos 5 g( ) cos What is the horizontal stretch shrink actor?. 5 What is the period o g()? What is the period o ()? b 5 What is the amplitude o ()? 4

30 Period b g( ) sin

31 Vertical and now horizontal stretch actors a: Vertical stretch, shrink actor b: horizontal stretch shrink actor. b ( ) asin b g( ) sin ( ) sin Relected across -ais. Vertically stretched by a actor o. Horizontally shrunk by a actor o 6.8 radians * 4 radians

32 Your turn: ( ) 5sin 3 g( ) sin Describe how () is a transormation o the parent unction g(). horizontal stretch actor. 3 g( ) sin ( ) 5sin 3 Relected across -ais. Vertically stretched by a actor o 5. Horizontally shrunk by a actor o 3 * radians 3 3

33 More transormations a. Vertical stretch, shrink actor g( ) sin ( ) asin b( c) k ( ) 3sin ( ) b: horizontal stretch Vertically stretched by a actor o 3. shrink actor. Horizontally stretched by a actor o b Shited right by π radians Not Relected across -ais. 6.8 radians * 4 radians

34 Your turn: g( ) Describe how the parent unction has been transormed. ( ) sin asin b( c) ( ) 3sin ( ) 3 3 k Relected across -ais. Vertically stretched by a actor o 3. Horizontally shrunk by a actor o Shited let by π/3 radians radians

35 Sinusoid. Amplitude: (½ o peak to peak distance) = ( ) asin b I a is negative: relection across -ais. Period: length o horizontal ais encompassing one complete cycle = b a 3. Frequency = b Frequency = /period

36 Sinusoid ( ) 3sin. Amplitude: (½ o peak to peak distance) = a 3. Period: length o horizontal ais encompassing one complete cycle = b 3. Frequency = b Frequency = /period

37 Your turn: ( ) What is the amplitude? (½ o peak to peak distance) = 9sin 3 a 9 What is the period o the unction? (length o horizontal ais encompassing one complete cycle) 3 * b What is the requency? 3 Frequency = /period 3 (radians) cycle cycle 3 radians = cycle or every 3 radians

38 Your turn: describe the transormations o () ( ) 0.5sin Vertical stretch/shrink 7. Horizontal stretch/shrink 8. Horizontal translation (phase shit) 9. Vertical translation