Consider the graph shown.

a)Explain why the function represented is periodic.

b) How many cycles are shown?

c) What are the maximum and minimum values?

d) What is the amplitude?

e) What is the period?

A shuttle bus takes passengers from a remote parking lot to the airport terminal 1.5 km away. The bus runs continually, completing a cycle in 10 min, which includes a 1-min stop at the parking lot and a 1—min stop at the terminal.

a) Sketch a graph to represent the position of the bus with respect to the parking lot as a function of time. Include two cycles.

b) What is the amplitude of the pattern?

c) Suppose that the bus increases its speed. Explain how this affects the graph.

Without using technology, sketch a graph of the sine function for values of x from -540° to 540°. Label the axes with an appropriate scale.

Without using technology, sketch a graph of the cosine function for values of x from -540° to 540°. Label the axes with an appropriate scale.

Consider the function y = cos (x + 60^o) + 3.

a) What is the amplitude?

b) What is the period?

c) Describe the phase shift.

d) Describe the vertical shift.

e) Graph the function for values of x from 0° to 360°.

f) How does the equation change if the phase shift and vertical shift are both in the opposite direction from the original function? Justify your answer.

A robot arm is used to cap bottles on an assembly line. The vertical position, y, in centimetres, of the arm after t seconds can be modelled by the function

\displaystyle y = 30\sin[360(t - 0.25)] + 45

a) Determine the amplitude, period, phase shift, and vertical shift.

b) What is the lowest vertical position that the arm reaches?

c) State the domain and range of the original sine function and the transformed function in set notation.

d) Suppose that the assembly line receives new bottles that require a lowest vertical position of 20 cm. How does the equation modelling the robot arm change?

Sunrise times for 1 year in Fort Erie, Ontario, measured on the last day of each month, starting in January, are shown using Eastern Standard Time (EST).

a) Convert the times to decimal format. Round each value to two decimal places. Create a scatter plot, with the month on the horizontal axis and the time on the vertical axis.

b) Construct a model of sunrise times using a sine function. State the amplitude, period, phase shift, and vertical shift.

c) Graph the model on the same axes as the data. Comment on the fit.

Use your model from question 7 to predict the sunrise time on January 7 and July 7.

The volume of blood in the left ventricle of an average-sized human heart varies from a minimum of about 50 mL to a maximum of about 130 mL. Ken has an average-sized heart and a resting pulse of 60 beats per minute. Assume that one heart beat represents the period.

a) Using a sine function, model the volume of blood in Ken’s left ventricle with respect to time.

b) Sketch a graph of volume in relation to time for four cycles.

c) How much blood is pumped from the left ventricle every minute?

d) How does the function change if you use a cosine function?

e) When Ken runs, his heart rate rises to 120 beats per minute. Adjust your model from part a] to reflect Ken’s heart rate while running.