Question

If the graph of y=cos⁡θ has a change in amplitude and a vertical translation, the equation becomes y=acos⁡θ+d, where a,d∈N and 0≤θ≤360∘. The graph of y=acos⁡θ+d is shown below.

The amplitude and the downward vertical translation, respectively, are:


6 and 2


7 and 1


3 and 4


3 and 2

Answer 1

The amplitude is 3 and the downward vertical translation is 4 ⇒ 3rd answer

Explanation:

* Lets revise some facts about the cosine function

- The Amplitude of cos(x) is the height from the center line to the

peak (or to the trough). Or we can measure the height from

highest to lowest points and divide that by 2.

- The Vertical Shift is how far the function is shifted vertically from

the usual position.

* Now lets solve the question

∵ y = cos(Ф)

- There is a change in amplitude, it becomes a

- There is a vertical translation by b units

∴ y = a cos(Ф) + d

* Now lets look to the graph to find a and d

- From the graph:

∵ The highest value is -1

∵ The lowest value is -7

∴ The amplitude a = (-1 - -7)/2 = (-1 + 7)/2 = 6/2 = 3

∵ The highest value of y = cos(Ф) is 1

∵ The amplitude is 3

∴ The highest value of y = acos(Ф) = 3

∵ The highest value of y = acos(Ф) + d is -1

∴ d = 3 - (-1) = 4 ⇒ means downward vertical translation by 4

∴ y = 3 cos(Ф) - 4

* The amplitude is 3 and the downward vertical translation is 4