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

Explanation:

Step1: Recall the cosine function transformation rules

The general form of a cosine function is (y = A\cos(B(x - C))+D). Here, (A) is the amplitude, (B) affects the period ((T=\frac{2\pi}{|B|})), (C) is the phase - shift ((C>0) shifts right, (C < 0) shifts left), and (D) is the vertical - shift. For (y=\cos(x)), when we consider (y = \cos(x - C)), the graph of (y=\cos(x)) is shifted (C) units to the right.

Step2: Use the point - substitution method

We know that for the black graph (y = \cos(x)), when (x = 0), (y=\cos(0)=1). For the red graph, when (x = 0), (y=-1).

• For (y=\cos(x)+1): When (x = 0), (y=\cos(0)+1=1 + 1=2).
• For (y=\cos(x-\pi)): Use the cosine subtraction formula (\cos(A - B)=\cos A\cos B+\sin A\sin B). So (y=\cos(x-\pi)=\cos x\cos\pi+\sin x\sin\pi). Since (\cos\pi=-1) and (\sin\pi = 0), then (y=-\cos x). When (x = 0), (y=-\cos(0)=-1).
• For (y=\cos(x)-2): When (x = 0), (y=\cos(0)-2=1-2=-1), but the shape of (y = \cos(x)-2) is a vertical shift of (y=\cos(x)) down by 2 units. The maximum value of (y=\cos(x)) is 1, and for (y=\cos(x)-2) the maximum value is (1 - 2=-1) and the minimum value is (-1-2=-3).
• For (y=\cos(x+\frac{\pi}{2})): Use the cosine addition formula (\cos(A + B)=\cos A\cos B-\sin A\sin B). So (y=\cos(x+\frac{\pi}{2})=\cos x\cos\frac{\pi}{2}-\sin x\sin\frac{\pi}{2}). Since (\cos\frac{\pi}{2}=0) and (\sin\frac{\pi}{2}=1), then (y=-\sin x). When (x = 0), (y=-\sin(0)=0).

Answer:

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