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Advanced Math

Fitting Sinusoidal Curves to Data

1. Use your graphing calculator to make a scatter plot of the given data.

2. Decide if you would like to model using a sine or cosine function; sometimes you are required to use both models.

3. Find the amplitude of the function. Recall:

4. Find the period (p). Recall that the period is the horizontal length to complete one cycle.

5. Find the value of B.

In radian mode: , therefore:

In degree mode: , therefore:

For modeling, set your calculator to DEGREE mode.

6. Find the vertical shift (the average of the highest and lowest data points), or

7. Find the phase shift. Locate the x value that corresponds with the first point of the cycle. This point will be different for sine and cosine. Set up a proportion:

Solve for h and determine if you require a right shift (–h) or a shift left (+h).

8. Write a sinusoidal model.

or

using the values obtained above.

9. Enter your sinusoidal function into your graphing calculator and determine if your model is a good fit. Make any adjustments as necessary.

Annual Temperature Change The table gives the average monthly temperature in

Montgomery County, Maryland.

|Month |(F | |Month |(F |

|Jan |40.0 | |Jul |85.8 |

|Feb |43.1 | |Aug |83.9 |

|Mar |54.6 | |Sep |76.9 |

|Apr |64.2 | |Oct |66.8 |

|May |73.8 | |Nov |55.5 |

|Jun |81.8 | |Dec |44.5 |

1. Use your graphing calculator to make a scatter plot of the given data. Sketch the graph below.

2. Decide if you would like to model using a sine or cosine function.

3. Find the amplitude, A, of the function. A =

4. Find the period, p. p =

5. Find the value of B. B =

6. Find the vertical shift, k. k =

7. Find the phase shift, h. h =

8. Write a sinusoidal model. y =

9. Enter your sinusoidal function into your graphing calculator y =

and determine if your model is a good fit.

Make any adjustments as necessary.

Annual Temperature Change The table gives the average monthly temperature in

Kansas City, Missouri.

|Month |(F | |Month |(F |

|Jan |28.8 | |Jul |79.1 |

|Feb |33.2 | |Aug |74.7 |

|Mar |43.1 | |Sep |64.8 |

|Apr |56.0 | |Oct |51.9 |

|May |68.3 | |Nov |39.7 |

|Jun |76.7 | |Dec |31.2 |

1. Use your graphing calculator to make a scatter plot of the given data. Sketch the graph below.

2. Decide if you would like to model using a sine or cosine function.

3. Find the amplitude, A, of the function. A =

4. Find the period, p. p =

5. Find the value of B. B =

6. Find the vertical shift, k. k =

7. Find the phase shift, h. h =

8. Write a sinusoidal model. y =

9. Enter your sinusoidal function into your graphing calculator y =

and determine if your model is a good fit.

Make any adjustments as necessary.

Height of a Tide The water depth in a narrow channel varies with the tides. The table shows the water depth over a 12-hour period.

|Time |Feet |

|12:00 a.m. |9.8 |

|1:00 a.m. |11.4 |

|2:00 a.m. |11.6 |

|3:00 a.m. |11.2 |

|4:00 a.m. |9.6 |

|5:00 a.m. |8.5 |

|6:00 a.m. |6.5 |

|7:00 a.m. |5.7 |

|8:00 a.m. |5.4 |

|9:00 a.m. |6.0 |

|10:00 a.m. |7.0 |

|11:00 a.m. |8.6 |

|12:00 p.m. |10.0 |

1. Use your graphing calculator to make a scatter plot of the given data. Sketch the graph below.

2. Decide if you would like to model using a sine or cosine function.

3. Find the amplitude, A, of the function. A =

4. Find the period, p. p =

5. Find the value of B. B =

6. Find the vertical shift, k. k =

7. Find the phase shift, h. h =

8. Write a sinusoidal model. y =

9. Enter your sinusoidal function into your graphing calculator y =

and determine if your model is a good fit.

Make any adjustments as necessary.

Predator/Prey When two species interact in a predator/prey relationship, the populations of both species tend to vary in a sinusoidal fashion. In a certain Midwestern county, the main food source for barn owls consists of field mice and other small mammals. The table gives the population of barn owls in this county every July 1 over a 12-year period.

|Year |Population |

|0 |50 |

|1 |52 |

|2 |73 |

|3 |80 |

|4 |71 |

|5 |60 |

|6 |51 |

|7 |43 |

|8 |29 |

|9 |20 |

|10 |28 |

|11 |41 |

|12 |49 |

1. Use your graphing calculator to make a scatter plot of the given data. Sketch the graph below.

2. Decide if you would like to model using a sine or cosine function.

3. Find the amplitude, A, of the function. A =

4. Find the period, p. p =

5. Find the value of B. B =

6. Find the vertical shift, k. k =

7. Find the phase shift, h. h =

8. Write a sinusoidal model. y =

9. Enter your sinusoidal function into your graphing calculator y =

and determine if your model is a good fit.

Make any adjustments as necessary.

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