1. Modeling with the Sine Function

Transcription

1 Name Date Sheilah Chason Intro to Sinusoidal Curves Math 433 Aim: To elore the stretching and shrinking of sinusoidal curves and to grah these curves with different amlitudes and eriods. 1. Modeling with the Sine Function Take a look at the following grah, which shows the aroimate average dail high temerature in New York's Central Park. # # Source: National Weather Service/The New York Times, Januar 7, 1996,. 36. Each ear, the attern reeats over and over, resulting in the following grah. Here, the -coordinate reresents time in ears with = 0 reresenting August 1, while the -coordinate reresents the temerature in o F. This is an eamle of cclical or eriodic behavior. Cclical behavior is common in the business world; just as there are seasonal fluctuations in the temerature in Central Park, there are seasonal fluctuations in the demand for surfing equiment, swimwear, snow shovels, and the list goes on. The following grah even suggests a cclical behavior in emloment at securities firms in the United States. Ý Net ear we will derive these tes of sinusoidal curves. For now we will elore changing the amlitude and frequenc.

2 Directions: Fill out the chart. Plot all oints. Do not connect the oints. Θ Deg. tanθ Y=tanθ Domain: Range:

3 Name Trig Grahs Date Sheilah Chason Math 433 Y=sinθ Domain: Range: Y=cosθ Domain: Range: Y= tanθ Domain: Range:

4 Amlitude: On a sinusoidal curve, the difference between the maimum and minimum oints divided b two. Θ Y=sinθ Y=sinθ Y=sinθ Y=-sinθ Y=cosθ Y=-cosθ Y=1.5cosθ -π -3 π/ - π - π/ 0 π/ π 3 π/ π Summar of Amlitude:

5 Frequenc: Θ Y=sinθ Y=sinθ Y=-sin θ Y=1.5sinθ Y=cosθ Y=cosθ Y=cosθ 0 π/4 π / 3 π/4 π 5 π/4 3 π/ 7 π/4 π

6 Θ Y=sinθ Y=sinθ Y=sinθ Y=-sinθ Y=cosθ Y=cosθ Y=-cosθ -π -3 π/ - π - π/ 0 π/ π 3 π/ π

7 Grah the following, over the given domain, indicating the amlitude, eriod and frequenc, and critical values. 1)Y=-1.5 sin; Amlitude: Frequenc: 0 π )Y=cos ; -π π Amlitude: Frequenc: 3)Y=sin; Amlitude: Frequenc: [-π,π]

8 1)Grah Y= -sin and Amlitude: Frequenc: Y= cos over the interval [0,π] Amlitude: Frequenc: )Grah Y= 1.5sin and Amlitude: Frequenc: Y= -cos over the interval [- π,π] Amlitude: Frequenc:

9 Grah the following, over the given domain, indicating the amlitude, eriod and frequenc, and critical values. 1)Y= sin; Amlitude: Frequenc: 0 π )Y=cos() ; Amlitude: Frequenc: [-π,π] 3)Y=-1.5sin(); [-π,π] Amlitude: Frequenc:

10 Grah both functions on the same coordinate ais and determine the number of oints of intersection. 1)Grah Y= sin and Amlitude: Frequenc: Y= cos over the interval [0,π] Amlitude: Frequenc: )Grah Y= -1.5sin and Amlitude: Frequenc: Y= cos over the interval [-π,π] Amlitude: Frequenc:

11 Grah the following, over the given domain, indicating the amlitude, eriod and frequenc, and critical values. 1)Y= tan; Frequenc: 0 π )Y=-tan ; [-π,π] Frequenc: 3)Y=-tan (); [-π,π] Frequenc:

12 Name Date Sheilah Chason Math 44 Aim: How to grah =asin(b)+c and =acos(b)+c. To Do: If sec 40 =1.3054, find the following: Cos 40 = Sec 140 = Sec -40 = Cos 0 = Name that Grah / / / 3/ / / / 3/ -3/ / / 3/ Domain: Domain: Domain: Range: Range: Range: Amlitude: Amlitude: Amlitude: / / / 3/ -3/ / / 3/ -0.4 / / Domain: Domain: Domain: Range: Range: Range: Amlitude: Amlitude: Amlitude: / / / 3/ -3/ / / 3/ -3/ / / 3/ Domain: Domain: Domain: Range: Range: Range:

13 Amlitude: Amlitude: Amlitude: Vertical Shift of main ais(c): G()=sin()+1 Amlitude: Frequenc: Vertical Shift: Domain: [-π,π] Range: G(π)= G()=sin()+1 Amlitude: Frequenc: Vertical Shift: Domain: [-π,π] Range: G(π)= f()=cos()- Amlitude: Frequenc: Vertical Shift: Domain: [0,π] Range:

14 F(π)= H()= -sin(3)- Amlitude: Frequenc: Vertical Shift: Domain: [-π/3,π/3] Range: H(-π) G()=cos()+3 Amlitude: Frequenc: Vertical Shift: Domain: [-π,π] Range: G(π/)= F() = -sin ()+ Amlitude: Frequenc: Vertical Shift:

15 Domain: [-π,π] Range: F(π)= M()=-.5sin()-1 Amlitude: Frequenc: Vertical Shift: Domain: [-π,π] Range: F(π)= S()= -1cos()- Amlitude: Frequenc: Vertical Shift: Domain: [-π,π] Range: s(π)= E()= sin3-4 Amlitude: Frequenc: Vertical Shift: Domain: [-π/3,π/3] Range:

16 s(π)= Sketch sin( ) and sin( ) 1 on the ais below. Sketch cos( ) and cos( ) on the ais below.

17 Grah the following, over the given domain, indicating the amlitude, eriod and frequenc, and critical values. 1)Y=-1.5 sin()-; Amlitude: Frequenc: 0 π )Y=cos()+1 ; Amlitude: Frequenc: -π π 3)Y=-sin()-3 [-π,π] Amlitude: Frequenc:

18 1)Grah and Y= over the interval [ ] Amlitude: Frequenc: Amlitude: Frequenc: ) Grah and Y= over the interval [ ] Amlitude: Frequenc: Amlitude: Frequenc: