Question

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

Explanation:

Step1: Analyze vertical shift

The black - graph \(y = \cos x\) has a maximum value of \(y = 1\) at \(x = 0\). The red - graph has a maximum value of \(y=3\) at \(x=\pi\). The vertical shift \(d\) of the cosine function \(y = A\cos(B(x - C))+D\) can be found by comparing the maximum values. The vertical shift from \(y = \cos x\) to the red graph is \(2\) units up, so \(D = 2\).

Step2: Analyze horizontal shift

The maximum of \(y=\cos x\) occurs at \(x = 0\), and the maximum of the red graph occurs at \(x=\pi\). For the cosine function \(y=\cos(x - C)\), when \(C=\pi\), the graph of \(y = \cos x\) is shifted \(\pi\) units to the right.

Step3: Rewrite the equation

Starting with the general form \(y - D=\cos(x - C)\), substituting \(D = 2\) and \(C=\pi\) gives \(y - 2=\cos(x-\pi)\). Also, we know that \(\cos(x-\pi)=-\cos x\) and \(\cos(x + \pi)=-\cos x\). So \(y - 2=\cos(x+\pi)\) is also correct.

Answer:

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