Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical trans

Question

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of the following function. Then graph the function over at least one period. y = -cos [2/3 (x - pi/3)] (a) The amplitude is 1 (b) The period is (Simplify your answer. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.) (c) The phase shift is (Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.) (d) The vertical translation is (e) The range is (Type your answer in interval notation.) Use the graphing tool to graph the function. Type pi to insert pi as needed.

Explanation / Answer

ANS: b)period for any derivative of cos function in the form of y = cos (bx- y) will always have the period = 2pi/b. Here b = 2/3 so the period is 2pi/2/3 = pi/3.

c) for phase shift if we write it in simplified form it is y = - cos (2/3x -2pi/9). Now we know that -cosx = cos (pi-x) using this equation can be written as y = cos (pi-(2/3x-2pi/9)) = cos (-2/3x - (-16pi/9)). So phase shift will be of -16pi/9 compared to y = cosx.

As there is nothing added to the equation there will be no vertical translation.

e) the range of y = cosx is {-1,1}. This function will have the same range as basically all the values for x gives the same value as cosx and there is no coefficient multiplied to cosx

for example if we multiply cosx by 3 then 3 cosx will have a range of {-3,3} but here the function is multiplied by -1 so the range remains the same.

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