Question

the black graph is $y = \cos x$. choose the equation for the red graph. $y = \cos(x + \pi) + 2$ $y = \cos(x) + 2$ $y = \cos(x + \pi/2) + 1$ $y = \cos(x - \pi/2)$

Explanation:

Step1: Analyze the vertical shift

The general form of a cosine function is \(y = A\cos(B(x - C))+D\), where \(D\) is the vertical shift. The black graph \(y = \cos x\) has a mid - line \(y = 0\), and the red graph has a mid - line \(y = 2\). So \(D=2\).

Step2: Analyze the horizontal shift

We know that for the function \(y=\cos(x + h)\), it is a horizontal shift of \(y = \cos x\). The point \((0,1)\) on \(y=\cos x\) is shifted. Let's check the function \(y=\cos(x+\pi)+2\). When \(x =-\pi\), \(y=\cos(-\pi+\pi)+2=\cos(0)+2=1 + 2=3\) (not relevant). For \(y=\cos(x)\), when \(x = 0\), \(y = 1\). For \(y=\cos(x+\pi)+2\), using the identity \(\cos(x+\pi)=-\cos x\).
The standard point \((0,1)\) of \(y = \cos x\):
For \(y=\cos(x+\pi)+2\), when \(x=-\frac{\pi}{2}\), \(y=\cos(-\frac{\pi}{2}+\pi)+2=\cos(\frac{\pi}{2})+2=0 + 2=2\)

Answer:

\(y=\cos(x+\pi)+2\)