which of the following is an equation for the graph? a. 3 sin(2x + π) - 1 b. 3 cos(2x + π) - 1 c. 3 sin(2x)…

Explanation:

Step1: Analyze the amplitude

The general form of a sinusoidal function is $y = A\sin(Bx - C)+D$ or $y = A\cos(Bx - C)+D$, where $|A|$ is the amplitude. From the graph, the distance from the maximum value to the mid - line is 3, so $|A| = 3$.

Step2: Analyze the vertical shift

The mid - line of the graph is $y=-1$. In the general form $y = A\sin(Bx - C)+D$ or $y = A\cos(Bx - C)+D$, $D$ represents the vertical shift. Here $D=-1$.

Step3: Analyze the period

The period $T$ of a sinusoidal function $y = A\sin(Bx - C)+D$ or $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. From the graph, the period $T=\pi$. Since $T = \pi=\frac{2\pi}{|B|}$, we get $|B| = 2$.

Step4: Analyze the phase - shift and function type

When $x = 0$, the function value is at a minimum. For a cosine function $y = A\cos(Bx - C)+D$, when $Bx - C=(2n + 1)\pi,n\in\mathbb{Z}$, the cosine function reaches its minimum. If we start with the basic cosine function $y=\cos x$ which has a minimum at $x=\pi$. For the function $y = A\cos(Bx - C)+D$ with $B = 2$, when $x = 0$, we want $2\times0 - C=\pi$, so $C=-\pi$. The function is $y = 3\cos(2x+\pi)-1$.

Answer:

B. $3\cos(2x+\pi)-1$