state the amplitude, period, frequency, phase shift and vertical shift of each function, Then graph two periods of the function of:
1) y=3 sin(x-pie/4)
2) y=0.25cosx+3
graphed by Wolfram
http://www.wolframalpha.com/input/?i=plot+y%3D3+sin%28x-π%2F4%29+%2C+y%3D0.25cos%28x%29+%2B3
you should be able to answer all your questions by looking at the graphs
To determine the amplitude, period, frequency, phase shift, and vertical shift of each function, we can use the general form of a sinusoidal function:
y = A*sin(B(x-C)) + D
1) Function: y = 3*sin(x-π/4)
Amplitude (A): The amplitude represents the maximum distance the graph of the function reaches from the centerline. In this case, the amplitude is 3.
Period (P): The period is the length of one complete cycle of the function. To determine the period, we can use the formula: P = 2π/B. In this function, B = 1, so the period is 2π.
Frequency (f): The frequency is the reciprocal of the period and represents the number of cycles per unit. In this case, f = 1/(2π) ≈ 0.159 Hz.
Phase Shift (C): The phase shift determines how the graph of the function is horizontally shifted. In this function, C = π/4, indicating a shift to the right by π/4 units.
Vertical Shift (D): The vertical shift determines how the graph of the function is vertically shifted. In this case, D = 0, indicating no vertical shift.
2) Function: y = 0.25*cos(x) + 3
Amplitude (A): The amplitude is 0.25, representing the maximum distance the graph reaches from the centerline.
Period (P): The period is determined by the coefficient of x, which is 1. Therefore, the period is 2π.
Frequency (f): The frequency is 1/period = 1/(2π) ≈ 0.159 Hz.
Phase Shift (C): There is no phase shift in this function, so C = 0.
Vertical Shift (D): The vertical shift is upward by 3 units, so D = 3.
Now, let's graph two periods of each function:
1) Graph of y = 3*sin(x-π/4):
To graph two periods, we will plot points from -2π to 4π.
2) Graph of y = 0.25*cos(x) + 3:
Again, we will plot points on the graph from -2π to 4π.
(Note: Due to text limitations, I am unable to provide you with an actual graph here. However, you can use any graphing tool or calculator to plot these functions and visually observe the graph.)