Which description matches the transformations y=cos x undergoes to produce y=-2cos3x?

A. horizontal compression by factor 3, vertical stretch by factor 2, then reflection across the x-axis.

B. horizontal shift left 2 units, then vertical shift up by 3 units.

C. reflection across the y-axis, vertical shift up by 2 units, horizontal shift right by 3 units.

D. horizontal stretch by factor 2, reflection across the x-axis, then vertical stretch by factor 3.

horizontal compression by factor 3, vertical stretch by factor 2, then reflection across the x-axis ⇒ answer A

Step-by-step explanation:

* Lets revise some transformation

- A vertical stretching is the stretching of the graph away from

the x-axis

# If k > 1, the graph of y = k•f (x) is the graph of f (x) vertically stretched

by multiplying each of its y-coordinates by k.

# If k should be negative, the vertical stretch is followed by a reflection

across the x-axis.

- A horizontal compression is the squeezing of the graph toward

the y-axis.

# If k > 1, the graph of y = f (k•x) is the graph of f (x) horizontally

compressed by dividing each of its x-coordinates by k.

* Lets solve the problem

∵ y = cos x

∵ y = - 2 cos 3x

- At first cos x multiplied by - 2

∵ y multiplied by - 2

∵ 2 > 1

∴ y = cos x is stretched vertically by factor 2

∵ The factor 2 is negative

∴ y = cos x reflected across the x-axis

∴ The function y = cos x stretched vertically with factor 2 and then

reflected across the x-axis ⇒ (1)

∵ cos x changed to cos 3x

∵ x multiplied by 3

∵ 3 > 1

∴ y = cos x compressed horizontally by factor 3

∴ The function y = cos x compressed horizontally by factor 3 ⇒ (2)

- From (1) and (2)

* The function y = cos x has horizontal compression by factor 3,

vertical stretch by factor 2, then reflection across the x-axis to

produce y = - 2 cos 3x