Final answer:
In summary, for the function f(x) = -1/3sin(2(x-45))-3, the amplitude is 1/3, the period is 180 degrees, the phase shift is 45 degrees to the right, and it has a vertical shift of 3 units downwards. Transformations include vertical reflection, vertical compression, horizontal compression, horizontal phase shift, and a vertical shift.
Step-by-step explanation:
To answer the student's question about the function f(x) = -\frac{1}{3}sin (2(x-45))-3, let's decipher it step by step:
a) Key Features for One Cycle of the Transformed Function
- The amplitude is \(\frac{1}{3}\), given by the coefficient in front of the sine function. Note that it is positive since amplitude is always non-negative.
- The period of the function is \(\frac{360^\circ}{2}\) or \(180^\circ\), due to the coefficient 2 included within the sine function which compresses the function horizontally. This means one cycle completes every \(180^\circ\).
- The phase shift is \(45^\circ\) to the right, indicated by the term \(x-45\) inside the function.
- The vertical shift is 3 units downwards, indicated by the -3 outside of the sine function.
b) Transformations Being Applied to the Base Function y = sinx
- Vertical reflection, since the coefficient in front of the sine is negative.
- Vertical compression by a factor of 1/3.
- Horizontal compression by a factor of 2.
- Horizontal phase shift 45 degrees to the right.
- Vertical shift 3 units down.
c) Sketch the Transformed Function
The student is expected to graph the function considering all the transformations mentioned. Starting with the base sine curve, reflect it vertically, compress it vertically by 1/3, compress it horizontally to have a period of 180 degrees, then shift it 45 degrees to the right, and finally shift it down by 3 units.