Question

which of the following graphs is the correct plot of ( y = - 4 sin ( x ) )?

Explanation:

Step1: Analyze the general form of the sine function

The general form of a sine function is \(y = A\sin(x)\), where \(A\) is the amplitude. For \(y=-4\sin(x)\), the amplitude \(|A| = 4\). This means the maximum value of \(y\) is \(4\) and the minimum value is \(-4\).

Step2: Consider the reflection

The negative sign in \(y =-4\sin(x)\) reflects the graph of \(y = 4\sin(x)\) over the \(x\) - axis.

Step3: Check key points

For \(y=\sin(x)\), when \(x = 0\), \(y = 0\); when \(x=\frac{\pi}{2}\approx1.57\), \(y = 1\); when \(x=\pi\approx3.14\), \(y = 0\); when \(x=\frac{3\pi}{2}\approx4.71\), \(y=-1\); when \(x = 2\pi\approx6.28\), \(y = 0\). For \(y=-4\sin(x)\), when \(x = 0\), \(y = 0\); when \(x=\frac{\pi}{2}\), \(y=-4\); when \(x=\pi\), \(y = 0\); when \(x=\frac{3\pi}{2}\), \(y = 4\); when \(x = 2\pi\), \(y = 0\).

Looking at the second graph (upper - right graph in the original four - graph set):

• At \(x = 0\), \(y=0\)
• As \(x\) increases from \(0\) to \(\frac{\pi}{2}\approx1.57\), \(y\) decreases to \(- 4\)
• As \(x\) increases from \(\frac{\pi}{2}\) to \(\pi\approx3.14\), \(y\) increases back to \(0\)
• As \(x\) increases from \(\pi\) to \(\frac{3\pi}{2}\approx4.71\), \(y\) increases to \(4\)
• As \(x\) increases from \(\frac{3\pi}{2}\) to \(2\pi\approx6.28\), \(y\) decreases back to \(0\)

Answer:

The upper - right graph (the second graph in the original four - graph set) is the correct plot of \(y=-4\sin(x)\)