Plot the 12 values on a graph. The number corresponding to the month is the first
coordinate of each point, and average temperature for that month is the second
coordinate. The points should create a periodic pattern. Assume that the data is, in
fact, periodic and use the graph to determine the following values for both a sine
function and a cosine function:
Sine Cosine
Vertical Shift
Horizontal Shift
Amplitude
Period
Domain
Range
Without specific data, it is not possible to plot the 12 values on a graph. However, we can discuss the general interpretation and characteristics of a sine and cosine function.
A sine function is defined as y = A*sin(B(x - C)) + D, where A represents the amplitude, B represents the reciprocal of the period, C represents the horizontal shift, and D represents the vertical shift.
A cosine function is defined as y = A*cos(B(x - C)) + D, where A represents the amplitude, B represents the reciprocal of the period, C represents the horizontal shift, and D represents the vertical shift.
Vertical Shift: The vertical shift, represented by D in both functions, indicates how much the function is shifted up or down. It is the value added or subtracted from the base function to obtain the desired shifting.
Horizontal Shift: The horizontal shift, represented by C in both functions, indicates how much the function is shifted to the left or right along the x-axis. A positive value of C indicates a shift to the right, while a negative value of C indicates a shift to the left.
Amplitude: The amplitude, represented by A in both functions, determines the maximum vertical distance between the curve and the horizontal axis. It represents half the total range of values the function takes.
Period: The period, represented by P or 2π/B (where B is the coefficient of x), represents the length of one complete cycle of the function. It is the horizontal length from one peak to the next, or one trough to the next.
Domain: The domain represents the set of all possible input values (x-values) for the function. In most cases, for a periodic function, the domain can be all real numbers unless there are specific restrictions.
Range: The range represents the set of all possible output values (y-values) for the function. It depends on the amplitude and vertical shift of the function.
To determine the values for the sine and cosine functions, we would require the specific data points or the actual values of the vertical shift, horizontal shift, amplitude, period, domain, and range.