Question

match the function with its graph.
$y=-2\sin(x)$

Explanation:

Step1: Analyze the general form of the sine function

The general form of a sine function is \(y = A\sin(Bx - C)+D\). For the function \(y=- 2\sin(x)\), we have \(A=-2\), \(B = 1\), \(C = 0\), \(D=0\).

Step2: Determine the amplitude

The amplitude of the function \(y = A\sin(x)\) is \(|A|\). Here, \(|A|=|-2| = 2\). This means the maximum value of \(y\) is \(2\) and the minimum value is \(-2\).

Step3: Analyze the reflection

Since \(A=-2<0\), the graph of \(y = \sin(x)\) is reflected about the \(x\) - axis.

Step4: Check the key points

For \(y=\sin(x)\), when \(x = 0\), \(y = 0\); when \(x=\frac{\pi}{2}\), \(y = 1\); when \(x=\pi\), \(y = 0\); when \(x=\frac{3\pi}{2}\), \(y=-1\); when \(x = 2\pi\), \(y = 0\). For \(y=-2\sin(x)\), when \(x = 0\), \(y = 0\); when \(x=\frac{\pi}{2}\), \(y=-2\); when \(x=\pi\), \(y = 0\); when \(x=\frac{3\pi}{2}\), \(y = 2\); when \(x = 2\pi\), \(y = 0\).

Answer:

The green - colored graph (the third graph in the image) is the graph of \(y=-2\sin(x)\)