Question

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

Explanation:

Step1: Analyze vertical shift

The black graph \(y = \cos x\) has a \(y -\)intercept at \((0,1)\). The red graph has a \(y -\)intercept at \((0,3)\).

Step2: Use vertical - shift formula

For a function \(y = f(x)+k\), the vertical shift is \(k\). If \(y=\cos x\) is shifted to \(y=\cos x + k\), when \(x = 0\), \(y=\cos(0)+k\). Since \(\cos(0)=1\) and \(y = 3\) (at \(x = 0\) for the red graph), then \(1 + k=3\), so \(k = 2\).

Answer:

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