Question

the black graph is $y = \sin x$. choose the equation for the red graph. $y = \sin(x) - 1$ $y = \sin(x + \pi)$ $y = \sin(x) + 1$ $y = \sin(x - \pi)$

Explanation:

Step1: Recall the transformation rules of sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\), where \(D\) represents vertical shift. If \(D> 0\), the graph shifts up; if \(D < 0\), the graph shifts down.

Step2: Analyze the given points

For \(y=\sin(x)\), when \(x = 0\), \(y=\sin(0)=0\). For the red - graph, when \(x = 0\), \(y = 1\).
Substitute \(x = 0\) into each function:

• For \(y=\sin(x)-1\), when \(x = 0\), \(y=\sin(0)-1=- 1\).
• For \(y=\sin(x +\pi)\), when \(x = 0\), \(y=\sin(\pi)=0\).
• For \(y=\sin(x)+1\), when \(x = 0\), \(y=\sin(0)+1=1\).
• For \(y=\sin(x-\pi)\), when \(x = 0\), \(y=\sin(-\pi)=0\).

Answer:

\(y=\sin(x)+1\)