Answer: \(a=3,\ b=2,\ c=-1\); amplitude \(2\); period \(\frac{2\pi}{3}\).

From the graph, the maximum value is \(2\) and the minimum value is \(-4\).

The midline is therefore

\(\frac{2+(-4)}{2}=-1,\)

so

\(c=-1.\)

The amplitude is

\(\frac{2-(-4)}{2}=3,\)

so

\(a=3.\)

The graph has consecutive maxima at \(x=0\) and \(x=\pi\), so the period is \(\pi\). For \(y=a\cos bx+c\), the period is

\(\frac{2\pi}{b}.\)

Thus

\(\frac{2\pi}{b}=\pi,\)

so

\(b=2.\)

For the curve \(y=2\sin3x+4\), the amplitude is

\(2.\)

The period is

\(\frac{2\pi}{3}.\)