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 sine - function properties

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $f(x)=\frac{5}{4}\sin(x)+1$, where $A = \frac{5}{4}$, $B = 1$, $C = 0$, and $D = 1$.

Step2: Find the maximum value

The maximum value of $\sin(x)$ is 1. Substitute $\sin(x)=1$ into $f(x)$: $f(x)=\frac{5}{4}\times1 + 1=\frac{5 + 4}{4}=\frac{9}{4}=2.25$.

Step3: Find the minimum value

The minimum value of $\sin(x)$ is - 1. Substitute $\sin(x)=-1$ into $f(x)$: $f(x)=\frac{5}{4}\times(-1)+1=\frac{-5 + 4}{4}=-\frac{1}{4}=-0.25$.

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

The derivative of $y = \sin(x)$ is $y'=\cos(x)$. For $x\in(0,\frac{\pi}{2})$, $\cos(x)>0$. Since $y = f(x)=\frac{5}{4}\sin(x)+1$ and the coefficient of $\sin(x)$ is positive ($\frac{5}{4}>0$), the function $f(x)$ is increasing on the interval $(0,\frac{\pi}{2})$.

Step5: Determine the range

The range of $\sin(x)$ is $[-1,1]$. For $y=\frac{5}{4}\sin(x)+1$, when $\sin(x)=-1$, $y =-\frac{1}{4}$; when $\sin(x)=1$, $y=\frac{9}{4}$. So the range is $[-\frac{1}{4},\frac{9}{4}]$.

Answer:

The maximum value of the function is $2.25$. The minimum value of the function is $-0.25$. On the interval $(0,\frac{\pi}{2})$, the function is increasing. The range of the function is $[-\frac{1}{4},\frac{9}{4}]$.