Question

17hot spot 1 pointselect the graph for the following function:$y = 2\sin\theta$

Explanation:

Step1: Recall the general form of a sine function

The general form of a sine function is \(y = A\sin(\theta)+B\). Here \(B = 0\) and \(A = 2\). The amplitude of the function \(y=\sin(\theta)\) is \(|A|\).

Step2: Determine the amplitude

For the function \(y = 2\sin(\theta)\), the amplitude \(|A|=|2| = 2\). The amplitude represents the maximum distance from the mid - line (\(y = 0\) in this case) to the peak (or trough) of the sine wave. The standard sine function \(y=\sin(\theta)\) has an amplitude of \(1\), and \(y = 2\sin(\theta)\) is a vertical stretch of \(y=\sin(\theta)\) by a factor of \(2\).

Step3: Analyze key points

When \(\theta=0\), \(y = 2\sin(0)=0\). When \(\theta=\frac{\pi}{2}\), \(y = 2\sin(\frac{\pi}{2})=2\). When \(\theta=\pi\), \(y = 2\sin(\pi)=0\). When \(\theta=\frac{3\pi}{2}\), \(y = 2\sin(\frac{3\pi}{2})=- 2\). When \(\theta = 2\pi\), \(y=2\sin(2\pi)=0\).

Answer:

The graph of \(y = 2\sin(\theta)\) is a sine wave with amplitude \(2\) (oscillating between \(y=-2\) and \(y = 2\)) and passing through the origin \((0,0)\). Among the given graphs, the one that has a maximum value of \(y = 2\), a minimum value of \(y=-2\) and passes through \((0,0)\) is the correct graph.