Respiratory Cycle For a person at rest, the velocity (in liters per second) of air flow

Fill in the blanks. One period of a sine or cosine function function is called one ________ of the sine curve or cosine curve.

Fill in the blanks. The ________ of a sine or cosine curve represents half the distance between the maximum and minimum values of the function.

Fill in the blanks. The period of a sine or cosine function is given by ________.

Fill in the blanks. For the function given by represents the ________ ________ of the graph of the function.

Fill in the blanks. For the function given by represents a ________ ________ of the graph of the function.

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 114, find the period and amplitude

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 1522, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 2326, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 2326, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 2326, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 2326, describe the relationship between the graphs of and Consider amplitude, period, and shifts.

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 2734, graph and on the same set of coordinate axes. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 3556, sketch the graph of the function. (Include two full periods.)

In Exercises 5762, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window.

In Exercises 5762, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window.

In Exercises 5762, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window.

In Exercises 5762, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window.

In Exercises 5762, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window.

In Exercises 5762, use a graphing utility to graph the function. Include two full periods. Be sure to choose an appropriate viewing window.

Graphical Reasoning In Exercises 6366, find and for the function such that the graph of matches the figure.

Graphical Reasoning In Exercises 6366, find and for the function such that the graph of matches the figure.

Graphical Reasoning In Exercises 6366, find and for the function such that the graph of matches the figure.

Graphical Reasoning In Exercises 6366, find and for the function such that the graph of matches the figure.

In Exercises 6770, find and for the function such that the graph of matches the figure.

In Exercises 6770, find and for the function such that the graph of matches the figure.

In Exercises 6770, find and for the function such that the graph of matches the figure.

In Exercises 6770, find and for the function such that the graph of matches the figure.

In Exercises 71 and 72, use a graphing utility to graph and in the interval Use the graphs to find real numbers such that

In Exercises 71 and 72, use a graphing utility to graph and in the interval Use the graphs to find real numbers such that

Respiratory Cycle For a person at rest, the velocity (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by where is the time (in seconds). (Inhalation occurs when and exhalation occurs when ) (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function. 7

Respiratory Cycle After exercising for a few minutes, a person has a respiratory cycle for which the velocity of air flow is approximated by where is the time (in seconds). (Inhalation occurs when and exhalation occurs when ) (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function. 7

Data Analysis: Meteorology The table shows the maximum daily high temperatures for Tallahassee and Chicago (in degrees Fahrenheit) for month with corresponding to January. (Source: National Climatic Data Center) (a) A model for the temperature in Tallahassee is given by Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data points and the model for the temperatures in Tallahassee. How well does the model fit the data? (c) Use a graphing utility to graph the data points and the model for the temperatures in Chicago. How well does the model fit the data? (d) Use the models to estimate the average maximum temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain. 7

Health The function given by approximates the blood pressure (in millimeters) of mercury at time (in seconds) for a person at rest. (a) Find the period of the function. (b) Find the number of heartbeats per minute. 7

Piano Tuning When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approximated by where is the time (in seconds). (a) What is the period of the function? (b) The frequency is given by What is the frequency of the note? 7

Data Analysis: Astronomy The percent of the moons face that is illuminated on day of the year 2007, where represents January 1, is shown in the table. (Source: U.S. Naval Observatory) (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. (c) Add the graph of your model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the moons percent illumination for March 12, 2007. 7

Fuel Consumption The daily consumption (in gallons) of diesel fuel on a farm is modeled by where is the time (in days), with corresponding to January 1. (a) What is the period of the model? Is it what you expected? Explain. (b) What is the average daily fuel consumption? Which term of the model did you use? Explain. (c) Use a graphing utility to graph the model. Use the graph to approximate the time of the year when consumption exceeds 40 gallons per day. 8

Ferris Wheel A Ferris wheel is built such that the height (in feet) above ground of a seat on the wheel at time (in seconds) can be modeled by (a) Find the period of the model. What does the period tell you about the ride? (b) Find the amplitude of the model. What does the amplitude tell you about the ride? (c) Use a graphing utility to graph one cycle of the model. True or False? 8

The graph of the function given by translates the graph of exactly one period to the right so that the two graphs look identical. 8

The function given by has an amplitude that is twice that of the function given by

The graph of is a reflection of the graph of in the -axis.

Writing Use a graphing utility to graph the function given by for several different values of and Write a paragraph describing the changes in the graph corresponding to changes in each constant. Conjecture 8

In Exercises 85 and 86, graph and on the same set of coordinate axes. Include two full periods. Make a conjecture about the functions.

In Exercises 85 and 86, graph and on the same set of coordinate axes. Include two full periods. Make a conjecture about the functions.

Exploration Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials and where is in radians. (a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Use a graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare? (c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added? 8

Exploration Use the polynomial approximations for the sine and cosine functions in Exercise 87 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain. (a) (b) (c) (d) (e) (f ) 8

In Exercises 8992, use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm.

In Exercises 8992, use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm.

In Exercises 8992, use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm.

In Exercises 8992, use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm.

In Exercises 9396, write the expression as the logarithm of a single quantity.

In Exercises 9396, write the expression as the logarithm of a single quantity.

In Exercises 9396, write the expression as the logarithm of a single quantity.

In Exercises 9396, write the expression as the logarithm of a single quantity.

Make a Decision To work an extended application analyzing the normal daily maximum temperature and normal precipitation in Honolulu, Hawaii, visit this texts website at college.hmco.com. (Data Source: NOAA)