Assignment Part 1: Write mathematical equations of sinusoids. 1. The following sinusoid is plotted below. Complete the following steps to model the curve using the cosine function.
a) What is the phase shift, h, of this curve? (2 points)
b) What is the vertical shift, k, of this curve? (2 points)
c) What is the amplitude, a, of this curve? (2 points)
d) What is the period and the frequency factor, b, of this curve? (2 points)

e) Write an equation using the cosine function that models this data set. (5 points)

2. The following points are a minimum and a maximum of a sinusoid.
Complete the following steps to model the curve using the sine function.
(4.5, 2), (1.5, 22)
a) What is the vertical shift, k, of this curve? (2 points)
b) What is the amplitude, a, of this curve? (2 points)
c) What is period and the frequency factor, b, of this curve? (2 points)
d) Write an equation using the sine function that models this curve. (5 points)

Part 2: Model real-world problems using sinusoids.
1. The average monthly high temperatures for Phoenix, Arizona from the years 1981-2010 are shown in the table below, with the value 1 representing the month of January, and the value of 12 representing the month of December. Complete the following steps to model the curve using the sine function. Month Average Temperature (°F)
1 69
2 72
3 78
4 86
5 95
6 104
7 106
8 104
9 100
10 89
11 76
12 67
a) Make a scatterplot of the data. Use the grid below or include a screenshot of the data plotted from a calculator. (2 points)

b) What is the minimum temperature in the data set? (1 point)
c) What is the maximum temperature in the data set? (1 point)
d) What is the vertical shift, k, for the data set? (2 points)
e) What is the amplitude, a, for the data set? (2 points)
f) What is the period and frequency factor for the data set? (2 points)
g) What is the phase shift for the data set? (2 points)

. h) Write an equation using the sine function that models this data set. (5 points)
i) When would you expect the temperature in Phoenix to be 82°? (2 points)

2. In the real-world, humans hear sound as an analog signal. This means the signal is a continuous waveform, which is completely processable by human ears. However, machines ‘hear’ sound a little differently, as they process sound digitally, which is a discrete waveform.

When sound is recorded or transmitted electronically, the continuous (analog) waveform is sampled to convert it to a discrete (digital) sequence. Sampling is the process of reducing a continuous-time signal to a discrete-time signal. As the sampling rate increases, the sound quality of the recording or transmission will improve.

The graphs below represent two different samples of a pure tone.

Sample 1 is taken 8 times per unit of time. Sample 2 is taken 16 times per unit of time. Sample 1: Sample 2:

a) Why would sample 2 reproduce a better replication of the pure signal? (2 points)

b) Write a sinusoid model for Sample 2. (5 points)