Question

Given the graph of $y=4\cos (x+\frac{\pi }{4})\forall x\in R.$ Select the correct statements for this function.

• A. Range: $(-5,5)$
• B. $y=0\text{at}x=\frac{-3\pi }{4},\frac{\pi }{4},\frac{5\pi }{4}\forall x\in [-\pi ,2\pi ]$
• C. Period: $2\pi$
• D. Period: $\pi$

Answer 1

Answer: (B), (C)

Period: $2\pi$

Given the graph of:
$y=4\cos (x+\frac{\pi }{4})$


The graph ranges from $-4\text{to}4$
$\Rightarrow -4\le y\le 4\Rightarrow y\in [-4,4]$
Or range of the function is $[-4,4]$

To find palces where $y=0\forall x\in [-\pi ,2\pi ]$
We need to look at the graph and identify points in the given region where the function cuts the x - axis.


From the graph, $y=0$ at $x=\frac{-3\pi }{4},\frac{\pi }{4},\frac{5\pi }{4}$

Now, using the fact that if $f\left(x\right)$ has a period $T.$
Then, $f(ax+b)$ will have a period $\frac{T}{|a|}.$

Using the same concept. we know $\cos x$ has a fundamental period $2\pi$
$\Rightarrow 4\cos (x+\frac{\pi }{4})$ will have a fundamental period $2\pi /1\text{or}2\pi$
Hence, $4\cos (x+\frac{\pi }{4})$ has a period of $2\pi .$