Question

which values of a, b, and c are possible?
( a = 6, b = 1, c = \frac { pi } { 3 } )
( a = 6, b = 3, c = pi )
( a = 3, b = 1, c = \frac { pi } { 3 } )
( a = 3, b = 6, c = pi )

Explanation:

Step1: Determine the amplitude \(a\)

The amplitude is the maximum distance from the mid - line to the peak. From the graph, the maximum value is \(6\) and the minimum is \(-6\). The formula for amplitude \(a=\frac{\text{Max}-\text{Min}}{2}\). So \(a = 6\).

Step2: Determine the period \(T\) and then \(b\)

The general form of a sinusoidal function is \(y = a\sin(bx + c)\) or \(y=a\cos(bx + c)\), and the period \(T=\frac{2\pi}{b}\). Looking at the graph, assume it is a cosine - like function (since it starts at a non - zero value). If we consider the standard form \(y = a\cos(bx + c)\), and assume no phase - shift for a moment (to check the period). The period of the function in the graph: if we count the number of cycles. Let's assume the function has a period \(T = 2\pi\) (by looking at the horizontal length of one full cycle). Using the formula \(T=\frac{2\pi}{b}\), if \(T = 2\pi\), then \(b = 1\).

Step3: Check the phase - shift (using the general form \(y=a\sin(bx + c)\) or \(y = a\cos(bx + c)\))

Let's assume the function is \(y=a\cos(bx + c)\). If \(a = 6\), \(b = 1\), and we can check the phase - shift. The standard cosine function \(y=\cos(x)\) has a maximum at \(x = 0\). If we assume a small phase - shift (by visual inspection, the graph is not shifted too much). For example, if \(y = 6\cos(x+\frac{\pi}{3})\), it still has an amplitude of \(6\) and a period of \(2\pi\) (since \(b = 1\)).

Answer:

\(a = 6,b = 1,c=\frac{\pi}{3}\) (the first option)