question graph exactly one cycle of the function f(x)=4cos(x). identify the maximum, minimum, and period of…

question graph exactly one cycle of the function f(x)=4cos(x). identify the maximum, minimum, and period of the function maximum: minimum: period:

question graph exactly one cycle of the function f(x)=4cos(x). identify the maximum, minimum, and period of the function maximum: minimum: period:

Answer

Explanation:

Step1: Recall cosine - function properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $f(x)=4\cos(x)$, we have $A = 4$, $B = 1$, $C = 0$, and $D = 0$.

Step2: Find the maximum value

The range of the basic cosine function $y=\cos(x)$ is $[- 1,1]$. When we multiply by $A = 4$, the range of $y = 4\cos(x)$ is $[-4,4]$. The maximum value of $y = 4\cos(x)$ occurs when $\cos(x)=1$. So, the maximum value is $4\times1 = 4$.

Step3: Find the minimum value

The minimum value of $y = 4\cos(x)$ occurs when $\cos(x)=-1$. So, the minimum value is $4\times(-1)=-4$.

Step4: Find the period

The period of the cosine function $y = A\cos(Bx - C)+D$ is given by the formula $T=\frac{2\pi}{|B|}$. Since $B = 1$, the period $T=\frac{2\pi}{|1|}=2\pi$.

Answer:

Maximum: $4$ Minimum: $-4$ Period: $2\pi$