question graph exactly one cycle of the function f(x)=5 cos(x). identify the maximum, minimum, and period of…

question graph exactly one cycle of the function f(x)=5 cos(x). identify the maximum, minimum, and period of the function maximum: 5 minimum: -5 period:

question graph exactly one cycle of the function f(x)=5 cos(x). identify the maximum, minimum, and period of the function maximum: 5 minimum: -5 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 $f(x)=5\cos(x)$, we have $A = 5$, $B = 1$, $C = 0$, and $D = 0$.

Step2: Find the period

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

Step3: Find the maximum and minimum

The amplitude of the function $y = A\cos(Bx - C)+D$ is $|A|$. Here, $A = 5$, so the amplitude is $|5| = 5$. For a cosine - function $y=\cos(x)$ which has a range of $[- 1,1]$, when we multiply by $A = 5$, the range of $y = 5\cos(x)$ is $[-5,5]$. The maximum value occurs when $\cos(x)=1$, so $y_{max}=5\times1 = 5$. The minimum value occurs when $\cos(x)=-1$, so $y_{min}=5\times(-1)=-5$.

Answer:

Maximum: 5 Minimum: - 5 Period: $2\pi$