Question

The graph of y=cos⁡x is transformed to y=a cos ⁡(x−c)+d by a vertical compression by a factor of 1/3 and a translation 2 units down. The new equation is:

y=3cos⁡x+2


y=1/3cos⁡(x−2)


y=1/3cos⁡x−2


y=1/3cos⁡x+2

Answer 1

The new equation is y = 1/3 cos(x) - 2 ⇒ 3rd answer

Explanation:

* Lets revise the trigonometry transformation

- If the equation is y = a cos(x - c) + d

# a is the scale factor of a vertical stretch or compression

# c is the phase shift (negative is to the right, positive is to the left)

# d is the vertical shift

- If y = cos(x)

∴ a = 1 , c = 0 , d = 0

* Now lets solve the problem

∵ There is a vertical compression by a factor of 1/3

∴ a = 1/3

∵ There is a translation 2 units down (vertical translation)

∴ d = -2

∵ There is now phase shift (horizontal translation)

∴ c = 0

* Now lets write the new equation

∴ y = 1/3 cos(x) - 2

* For more understand look to the attached color graph

- The red is y = cos(x)

- The blue is y = 1/3 cos(x) - 2

Answer 2

`y = (1)/(3) \cos(x) - 2`

EXPLANATION

If the graph of y=cos⁡x is transformed to y=a cos ⁡(x−c)+d by a vertical compression by a factor of 1/3 and a translation 2 units down,

then

a=1/3

and d=-2.

The 'c' is a phase shift since it is not given, it means it is zero.

Therefore the new equation is:

y=1/3cos⁡(x-0)−2

This simplifies to:

y=1/3cos⁡x−2

The correct option is C.