A periodic function that does not start at the sinusoidal axis or at a maximum or a minimum has been shifted horizontally. This horizontal movement invites different people to see different starting points since a sine wave does not have a beginning or an end.

What are five other ways of writing the function

#### Watch This

http://www.youtube.com/watch?v=wUzARNIkH-g James Sousa: Graphing Sine and Cosine with Various Transformations

#### Guidance

The general sinusoidal function is:

The constant

Generally

**Example A**

Graph the following function:

**Solution: ** First find the start and end of one period and sketch only that portion of the sinusoidal axis. Then plot the 5 important points for a cosine graph while keeping the amplitude in mind.

**Example B**

Given the following graph, identify equivalent sine and cosine algebraic models.

**Solution:** Either this is a sine function shifted right by

**Example C**

At

**Solution: ** Since the period is 60 which works extremely well with the

Time (minutes) |
Height (feet) |

5 | 2 |

20 | 42 |

35 | 82 |

50 | 42 |

65 | 2 |

William chooses to see a negative cosine in the graph. He identifies the amplitude to be 40 feet. The vertical shift of the sinusoidal axis is 42 feet. The horizontal shift is 5 minutes to the right.

The period is 60 (not 65) minutes which implies

Thus one equation would be:

**Concept Problem Revisited**

The function

It all depends on where you choose start and whether you see a positive or negative sine or cosine graph.

#### Vocabulary

** Phase shift** is a typical horizontal shift left or right that is used primarily with periodic functions.

#### Guided Practice

1. Tide tables report the times and depths of low and high tides. Here is part of tide report from Salem, Massachusetts dated September 19, 2006.

10:15 AM |
9 ft. |
High Tide |

4:15 PM |
1 ft. |
Low Tide |

10:15 PM |
9 ft. |
High Tide |

Find an equation that predicts the height based on the time. Choose when

2. Use the equation from Guided Practice #1 to predict the height of the tide at 6:05 AM.

3. Use the equation from Guided Practice #1 to find out when the tide will be at exactly 8 ft on September

**Answers:**

1. There are two logical places to set

Time (hours : minutes) |
Time (minutes) |
Tide (feet) |

10:15 | 615 | 9 |

16:15 | 975 | 1 |

22:15 | 1335 | 9 |

5 | ||

5 |

These numbers seem to indicate a positive cosine curve. The amplitude is four and the vertical shift is 5. The horizontal shift is 615 and the period is 720.

Thus one equation is:

2. The height at 6:05 AM or 365 minutes is:

3. This problem is slightly different from question 2 because instead of giving

There are four times within the 24 hours when the height is exactly 8 feet. You can convert these times to hours and minutes if you prefer.

#### Practice

Graph each of the following functions.

1.

2.

3.

4.

5.

Give one possible sine equation for each of the graphs below.

6.

7.

8.

Give one possible cosine function for each of the graphs below.

9.

10.

11.

The temperature over a certain 24 hour period can be modeled with a sinusoidal function. At 3:00, the temperature for the period reaches a low of

12. Find an equation that predicts the temperature based on the time in minutes. Choose

13. Use the equation from #12 to predict the temperature at 4:00 PM.

14. Use the equation from #12 to predict the temperature at 8:00 AM.

15. Use the equation from #12 to predict the time(s) it will be

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 5.6.