Question

The graph of $f\left(x\right)=a\cos (bx+c)$ with $b>0$ is shown below. Then choose the correct option(s) for the given function?

• A. Period of $f\left(x\right)$ is $\frac{2\pi }{3}$
• B. $a=-1$
• C. $b=3$
• D. $a=1$

Answer 1

Answer: (A), (B), (C), (D)

$a=1$

There are two zeros that delimit half a cycle.
Hence, half a period is equal to:
$-\frac{\pi }{8}-(-\frac{11\pi }{24})=\frac{\pi }{3}$
Therefore, period of $f\left(x\right)$ is
$P=2\times \frac{\pi }{3}=\frac{2\pi }{3}$
But,period of $f\left(x\right)=a\cos (bx+c)$ is $\frac{2\pi }{|b|}$
So, $\frac{2\pi }{3}=\frac{2\pi }{|b|}$
$\Rightarrow b=\pm 3$
$\Rightarrow b=3(\because b>0)$

The range of $f\left(x\right)=a\cos (bx+c)$ is $[-a,a]$
From graph, the range of $f\left(x\right)=a\cos (bx+c)$ is $[-1,1]$
So, $a=\pm 1$