question graph exactly one cycle of the function f(x)=3 cos(2x). identify the maximum, minimum, and period…

Explanation:

Step1: Recall cosine - function properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $f(x)=3\cos(2x)$, where $A = 3$, $B = 2$, $C = 0$, and $D = 0$.

Step2: Find the maximum value

The range of the basic cosine function $y=\cos(t)$ is $[- 1,1]$. When we have $y = A\cos(Bx)$, the amplitude is $|A|$. Here $A = 3$, so the maximum value of $y = 3\cos(2x)$ is $|A|=3$.

Step3: Find the minimum value

Since the minimum value of $\cos(2x)$ is $-1$, when we multiply by $A = 3$, the minimum value of $y = 3\cos(2x)$ is $-|A|=-3$.

Step4: Find the period

The period of the cosine function $y=\cos(Bx)$ is given by the formula $T=\frac{2\pi}{|B|}$. Here $B = 2$, so $T=\frac{2\pi}{2}=\pi$.

Answer:

Maximum: $3$ Minimum: $- 3$ Period: $\pi$