Question

25. To produce the graph of the function y=0.5cos(0.5x), what transformations should be applied to the graph of the parent function y=cot(x)?

a. a horizontal compression to produce a period of pi/2 and a vertical compression
b. a horizontal compression to produce a period of pi/2 and a vertical stretch
c. a horizontal stretch to produce a period of 2pi and a vertical compression
d. a horizontal stretch to produce a period of 2pi and a vertical stretch

Answer 1

C is correct...

Explanation:

just took it on edge

Answer 2

C. A horizontal stretch to produce a period of
`2\pi` and a vertical compression.

Explanation:

We are given the parent function as
`y= \cot x`

It is given that, transformations are applied to the parent function in order to obtain the function
`y=0.5\cot (0.5x)` i.e.
`y=(1)/(2)\cot ((x)/(2))`

That is, we see that,

The parent function
`y= \cot x` is stretched horizontally by the factor of
`(1)/(2)` which gives the function
`y=\cot ((x)/(2))`.

Further, the function is compressed vertically by the factor of
`(1)/(2)` which gives the function
`y=(1)/(2)\cot ((x)/(2))`.

Now, we know,

If a function f(x) has period P, then the function cf(bx) will have period
`(P)/(|b|)`.

Since, the period of
`y= \cot x` is
`\pi`, so the period of
`y=(1)/(2)\cot ((x)/(2))` is
`(\pi)/(1/2)` =
`2\pi`

Hence, we get option C is correct.