Question

what is the value of b in the function $f(x) = \cos(bx)$ in the graph?
(1 point)
\\( \circ f(x) = \sin\left(-\frac{3x}{2}\
ight) \\)
\\( \circ f(x) = \sin\left(\frac{3x}{2}\
ight) \\)
\\( \circ f(x) = \cos\left(\frac{3x}{2}\
ight) \\)
\\( \circ f(x) = \cos\left(\frac{2x}{3}\
ight) \\)

Explanation:

Step1: Recall the period formula

The period formula for \(y = A\cos(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\).

Step2: Determine the period from the graph

From the graph, the period \(T=\frac{4\pi}{3}\).

Step3: Solve for \(B\)

Using the period formula \(T = \frac{2\pi}{|B|}\), substitute \(T=\frac{4\pi}{3}\). Then \(\frac{4\pi}{3}=\frac{2\pi}{|B|}\). Cross - multiply gives \(4\pi|B|=6\pi\). Divide both sides by \(2\pi\), we get \(2|B| = 3\), so \(|B|=\frac{3}{2}\).

Answer:

\(f(x)=\cos(\frac{3x}{2})\)