the graph of the function f(x) = 5/4 sin(x) + 1 is shown. what are the key features of this function? the…

Explanation:

Step1: Recall the range of sine function

The range of $y = \sin(x)$ is $[- 1,1]$.

Step2: Find the maximum value

For $y=\frac{5}{4}\sin(x)+1$, when $\sin(x) = 1$, $y=\frac{5}{4}\times1 + 1=\frac{5 + 4}{4}=\frac{9}{4}$.

Step3: Find the minimum value

When $\sin(x)=-1$, $y=\frac{5}{4}\times(-1)+1=\frac{-5 + 4}{4}=-\frac{1}{4}$.

Step4: Analyze the interval $(0,\frac{\pi}{2})$

The derivative of $y = \frac{5}{4}\sin(x)+1$ is $y'=\frac{5}{4}\cos(x)$. On the interval $(0,\frac{\pi}{2})$, $\cos(x)>0$, so $y'>0$ and the function is increasing.

Step5: Determine the range

Since the minimum value is $-\frac{1}{4}$ and the maximum value is $\frac{9}{4}$, the range is $[-\frac{1}{4},\frac{9}{4}]$.

Answer:

The maximum value of the function is $\frac{9}{4}$. The minimum value of the function is $-\frac{1}{4}$. On the interval $(0,\frac{\pi}{2})$, the function is increasing. The range of the function is $[-\frac{1}{4},\frac{9}{4}]$.