14. A summer resort town often shows
seasonal variations in the percent of the
workforce employed. The table lists the
percent employed on the first of each
month, starting in January, for 1 year.
Month | Employment (%)
1 | 62
2 | 67
3 | 75
4 | 80
5 | 87
6 | 92
7 | 96
8 | 93
9 | 89
10 | 79
11 | 72
12 | 65
a) Construct a model for these data using
either a sine function or a cosine
function. State the amplitude, period,
phase shift, and vertical shift.
How would I get the amplitude, period,
phase shift, and vertical shift values?
To determine the amplitude, period, phase shift, and vertical shift values for the given data, you can follow these steps:
1. Amplitude: The amplitude of a sine or cosine function represents the maximum deviation from the average. In this case, you can find the amplitude by calculating half the difference between the highest and lowest employment percentages. The formula for amplitude is:
Amplitude = (Highest Value - Lowest Value) / 2
From the data provided, the highest value is 96, and the lowest value is 62. Plugging these values into the formula:
Amplitude = (96 - 62) / 2 = 34 / 2 = 17
So the amplitude is 17.
2. Period: The period of a sine or cosine function represents the length of one complete cycle. In this case, you can determine the period by finding the number of months it takes for the employment percentage to repeat. Since the data covers 12 months, the period can be determined as:
Period = Number of Months in the Data
In this case, the period is 12 months.
3. Phase Shift: The phase shift for a sine or cosine function represents the horizontal displacement of the graph. It indicates the delay or advance of the function compared to the standard sine or cosine curve. To find the phase shift, you need to determine the month where the function begins. In this case, the function starts in January (month 1), so the phase shift is:
Phase Shift = Month When the Function Starts - 1
In this case, the phase shift is 1 - 1 = 0.
4. Vertical Shift: The vertical shift represents the vertical displacement of the function. It indicates the shift of the entire graph up or down. To find the vertical shift, you need to determine the average of the highest and lowest employment percentages. The formula for the vertical shift is:
Vertical Shift = (Highest Value + Lowest Value) / 2
From the data provided, the highest value is 96, and the lowest value is 62. Plugging these values into the formula:
Vertical Shift = (96 + 62) / 2 = 158 / 2 = 79
So the vertical shift is 79.
By applying these steps, you can determine the amplitude, period, phase shift, and vertical shift values for the given data.