what is the general equation of a cosine function with an amplitude of 3, a period of 4π, and a horizontal…

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine - function is $y = A\cos(B(x - C))+D$, where $A$ is the amplitude, $B$ is related to the period by $T=\frac{2\pi}{B}$, $C$ is the horizontal shift, and $D$ is the vertical shift (in this case $D = 0$).

Step2: Determine the value of $A$

Given that the amplitude $A = 3$.

Step3: Determine the value of $B$

We know that the period $T = 4\pi$, and since $T=\frac{2\pi}{B}$, we substitute $T = 4\pi$ into the formula: $4\pi=\frac{2\pi}{B}$. Solving for $B$, we cross - multiply to get $4\pi B=2\pi$, then $B=\frac{2\pi}{4\pi}=0.5$.

Step4: Determine the value of $C$

Given that the horizontal shift is $-\pi$, so $C=-\pi$.

Step5: Write the equation of the cosine function

Substitute $A = 3$, $B = 0.5$, and $C=-\pi$ into the general form $y = A\cos(B(x - C))$. We get $y = 3\cos(0.5(x-(-\pi)))=3\cos(0.5(x + \pi))$.

Answer:

$y = 3\cos(0.5(x+\pi))$ (the third option)