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

Answer:

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

Step-by-step 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. $y=\frac{1}{2}\cot (\frac{x}{2})$

That is, we see that,

The parent function $y= \cot x$ is stretched horizontally by the factor of $\frac{1}{2}$ which gives the function $y=\cot (\frac{x}{2})$.

Further, the function is compressed vertically by the factor of $\frac{1}{2}$ which gives the function $y=\frac{1}{2}\cot (\frac{x}{2})$.

Now, we know,

If a function f(x) has period P, then the function cf(bx) will have period $\frac{P}{|b|}$.

Since, the period of $y= \cot x$ is $\pi$, so the period of $y=\frac{1}{2}\cot (\frac{x}{2})$ is $\frac{\pi}{1/2}$ = $2\pi$

Hence, we get option C is correct.