Question

review the graph. which function represents the graph? ( y = - 3 cos ( x ) ) ( y = - 3 sin ( x ) ) ( y = 3 cos ( x ) ) ( y = 3 sin ( 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. The amplitude \(|A|\) is the maximum distance from the mid - line (here \(y = 0\)) to the graph.

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

For \(y=-3\cos(x)\), when \(x = 0\), \(y=-3\cos(0)=- 3\).
For \(y=-3\sin(x)\), when \(x = 0\), \(y=-3\sin(0)=0\).
For \(y = 3\cos(x)\), when \(x = 0\), \(y=3\cos(0)=3\).
For \(y = 3\sin(x)\), when \(x = 0\), \(y=3\sin(0)=0\).
The graph passes through the origin \((0,0)\), so we can eliminate \(y=-3\cos(x)\) and \(y = 3\cos(x)\) since they do not have \(y = 0\) when \(x = 0\).

Step3: Analyze the slope at \(x = 0\)

The derivative of \(y=-3\sin(x)\) is \(y^\prime=-3\cos(x)\), and \(y^\prime(0)=-3\cos(0)=-3\) (negative slope at \(x = 0\)).
The derivative of \(y = 3\sin(x)\) is \(y^\prime=3\cos(x)\), and \(y^\prime(0)=3\cos(0)=3\) (positive slope at \(x = 0\)).
Looking at the graph, as \(x\) increases from \(0\), \(y\) increases. So the slope at \(x = 0\) is positive.

Answer:

\(y = 3\sin(x)\)