The function provided is a sinusoidal function, which has its characteristic shape because of the sine function.
In the given function, f(x) = -11sin(7/3x + 1/6) - 2, the -2 represents an overall vertical shift in the function. This shifts the midline, or the line about which the function oscillates, down by 2 units.
The -11 in the function is the amplitude of the function, which determines the height of the peak and the depth of the trough of the sinusoidal function.
The range, or the set of possible output values, is determined by the amplitude and the overall shift. The range will extend from the shift minus the amplitude to the shift plus the amplitude.
As the shift is -2 and amplitude is -11, the range is calculated as followed:
To find the lower limit of the range, subtract the amplitude from the shift. Hence, the lower limit is -2 - (-11).
This results in a lower limit of 9 degrees Fahrenheit.
Next, to find the upper limit of the range, add the amplitude to the shift. Hence, the upper limit is -2 + (-11).
This leads to an upper limit of -13 degrees Fahrenheit.
Therefore, the average rate of change in temperature falls in the range of -13 degrees Fahrenheit to 9 degrees Fahrenheit.