Question

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

Explanation:

Step1: Recall the transformation rules

For a function \(y = f(x)\), a horizontal shift of \(h\) units is \(y=f(x - h)\) (right shift if \(h>0\), left shift if \(h < 0\)) and a vertical shift of \(k\) units is \(y=f(x)+k\) (up shift if \(k>0\), down shift if \(k < 0\)).

Step2: Analyze the key - point transformation

The key point of \(y = \cos x\) is \((0,1)\). For the red graph, the key point is \((\frac{\pi}{2},2)\).
First, consider the horizontal transformation: from \(x = 0\) to \(x=\frac{\pi}{2}\), we have a shift of \(x\to x-\frac{\pi}{2}\) (a right - shift of \(\frac{\pi}{2}\) units).
Second, consider the vertical transformation: from \(y = 1\) to \(y = 2\), we have a shift of \(y\to y - 1\) (an up - shift of \(1\) unit).
Substitute \(x\) with \(x-\frac{\pi}{2}\) and \(y\) with \(y - 1\) in the equation \(y=\cos x\).
We get \(y - 1=\cos(x-\frac{\pi}{2})\).

Answer:

\(y - 1=\cos(x-\frac{\pi}{2})\)