Question

identify the function displayed by the graph. (1 point) \\(\bigcirc f(x) = 6\sin x\\) \\(\bigcirc f(x) = 8\sin x\\) \\(\bigcirc f(x) = -6\sin x\\) \\(\bigcirc f(x) = 6\cos x\\)

Explanation:

Step1: Recall the general form of sine and cosine functions

The general form of a sine function is \(y = A\sin(x)\) and for a cosine function is \(y=A\cos(x)\), where \(A\) is the amplitude (\(\vert A\vert=\text{maximum value}-\text{mid - line value}\)). The mid - line of the given graph is \(y = 0\). The maximum value (in magnitude) of the function is \(6\). So, \(\vert A\vert=6\).

Step2: Check the value of the function at \(x = 0\)

For \(y = A\sin(x)\), when \(x = 0\), \(y=A\sin(0)=0\). For \(y = A\cos(x)\), when \(x = 0\), \(y = A\cos(0)=A\). In the given graph, when \(x = 0\), \(y = 0\), so the function is a sine function.

Step3: Determine the sign of \(A\)

We know that \(y=\sin(x)\) has a positive slope at \(x = 0\) (\(y^\prime=\cos(x)\), and \(\cos(0) = 1\)). The given graph has a negative slope at \(x = 0\). The derivative of \(y=-6\sin(x)\) is \(y^\prime=-6\cos(x)\), and at \(x = 0\), \(y^\prime=- 6\).

Answer:

\(f(x)=-6\sin x\)