graph exactly one cycle of the function $f(x)=-5cos(\frac{3}{4}x)+3$. identify the maximum, minimum, and…

Explanation:

Step1: Recall cosine - function properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $f(x)=- 5\cos(\frac{3}{4}x)+3$, we have $A=-5$, $B = \frac{3}{4}$, $C = 0$, and $D = 3$.

Step2: Find the maximum value

The range of the cosine function $\cos(\frac{3}{4}x)$ is $[-1,1]$. When $\cos(\frac{3}{4}x)=-1$, $f(x)=-5\times(-1)+3$. $f(x)=5 + 3=8$.

Step3: Find the minimum value

When $\cos(\frac{3}{4}x)=1$, $f(x)=-5\times1 + 3$. $f(x)=-5 + 3=-2$.

Step4: Find the period

The period formula for a cosine function $y = A\cos(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Since $B=\frac{3}{4}$, then $T=\frac{2\pi}{\frac{3}{4}}=\frac{8\pi}{3}$.

Answer:

Maximum: $8$ Minimum: $-2$ Period: $\frac{8\pi}{3}$