Question

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

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 $y = 2$. This indicates a vertical shift of 1 unit upwards. The general form of a vertical shift of a function $y = f(x)$ is $y=f(x)+k$, where $k$ is the amount of vertical shift. Here, $k = 1$.

Step2: Analyze horizontal shift

The black - graph $y=\cos x$ has a maximum at $x = 0$. The red - graph has a maximum at $x=\frac{\pi}{2}$. For the cosine function $y = \cos(x - h)$, a positive $h$ value shifts the graph to the right. When $h=\frac{\pi}{2}$, the graph of $y=\cos(x)$ is shifted $\frac{\pi}{2}$ units to the right.

Step3: Determine the equation

Combining the horizontal shift of $\frac{\pi}{2}$ units to the right and the vertical shift of 1 unit upwards, the equation of the red - graph is $y=\cos(x-\frac{\pi}{2}) + 1$.

Answer:

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