Question

the black graph is y = cos x. choose the equation for the red graph. y = cos(x + π) + 1 y = cos(x - π) - 1 y = cos(x + π) + 2 y = cos(x - π) - 2

Explanation:

Step1: Recall the transformation rules

The general form of a cosine function transformation is \(y = A\cos(B(x - C))+D\), where \(C\) is the phase - shift (horizontal shift), \(D\) is the vertical shift.

Step2: Analyze the phase - shift

For the function \(y=\cos x\), when we consider the point \((0,1)\) on \(y = \cos x\). For the red graph, if we substitute \(x=\pi\) into \(y=\cos(x - C)-D\). Using the identity \(\cos(x-\pi)=-\cos x\).
If we take the function \(y = \cos(x-\pi)-1\), when \(x = \pi\), \(y=\cos(\pi-\pi)-1=\cos(0)-1=1 - 1=-1\)

Step3: Analyze the vertical shift

The mid - line of \(y=\cos x\) is \(y = 0\). The mid - line of the red graph:
For \(y=\cos(x-\pi)-1\), since \(\cos(x-\pi)=-\cos x\), the range of \(y = \cos(x-\pi)-1\) is \([-1 - 1,1 - 1]=[-2,0]\) (because the range of \(\cos t\) is \([-1,1]\), for \(t=x - \pi\), \(\cos(x-\pi)\) has range \([-1,1]\) and then subtract \(1\)).

Answer:

\(y=\cos(x - \pi)-1\)