Find the vertical/horizontal shift, amplitude, period, domain, and range of the given values. Find the sine/cosine functions.
(1,67) (2,71) (3,78) (4,86) (5,95) (6,104) (7,106) (8,105) (9,100) (10, 88) (11, 76) (12,56)
To find the vertical/horizontal shift, amplitude, period, domain, and range of the given values, we need to analyze the pattern and characteristics of the data points.
Looking at the given values (1,67), (2,71), (3,78), (4,86), (5,95), (6,104), (7,106), (8,105), (9,100), (10,88), (11,76), and (12,56), we can see that they represent a set of points on a graph of a function.
First, we need to identify whether the given points follow a sine function or a cosine function. We can do this by determining if the y-values (67, 71, 78, etc.) correspond to the amplitude of the function or if they represent values shifted up or down from the axis.
To do this, we can calculate the average of the maximum and minimum values of the y-coordinates:
Maximum value: 106
Minimum value: 56
(106 + 56) / 2 = 81
The average of the maximum and minimum values is 81. This means that the vertical shift of the function is 81 units. Since the maximum value is at (7, 106), the function reaches its maximum value at x = 7.
From this information, we can deduce that the function is a cosine function, as it reaches its maximum value at x = 7, which corresponds to the peak of a cosine graph.
Now let's analyze the characteristics of the cosine function:
1. Vertical Shift: The vertical shift is given by the average of the maximum and minimum values, which is 81. This means that the function is shifted up by 81 units.
2. Amplitude: The amplitude is half the distance between the maximum and minimum values. In this case, it is (106 - 56) / 2 = 25. This means that the amplitude is 25.
3. Period: The period of a cosine function is determined by the length of one full cycle. From the given points, we can see that the function completes one full cycle from x = 1 to x = 12. Therefore, the period is 12 - 1 = 11.
4. Domain: The domain is the set of possible x-values for the function. In this case, we have the points from x = 1 to x = 12, so the domain is [1, 12].
5. Range: The range is the set of y-values that the function takes. From the given points, we can see that the y-values range from 56 to 106, so the range is [56, 106].
In summary:
Sine Function: N/A
Cosine Function:
- Vertical Shift: 81 units up
- Amplitude: 25
- Period: 11
- Domain: [1, 12]
- Range: [56, 106]