select the correct answer from each drop - down menu.\nthe graph of the function $f(x)=\frac{5}{4}sin(x)+1$…

select the correct answer from each drop - down menu.\nthe graph of the function $f(x)=\frac{5}{4}sin(x)+1$ is shown. what are the key features of this function?\nthe maximum value of the function is\nthe minimum value of the function is\non the interval $(0,\frac{pi}{2})$, the function is\nthe range of the function is\nreset next

select the correct answer from each drop - down menu.\nthe graph of the function $f(x)=\frac{5}{4}sin(x)+1$ is shown. what are the key features of this function?\nthe maximum value of the function is\nthe minimum value of the function is\non the interval $(0,\frac{pi}{2})$, the function is\nthe range of the function is\nreset next

Answer

Explanation:

Step1: Determine amplitude and vertical shift

Amplitude is (\frac{5}{4}), vertical shift is (+1).

Step2: Calculate maximum value

Maximum of (\sin(x)) is (1), so (f(x)_{\text{max}} = \frac{5}{4}(1) + 1 = \frac{9}{4} = 2.25).

Step3: Calculate minimum value

Minimum of (\sin(x)) is (-1), so (f(x)_{\text{min}} = \frac{5}{4}(-1) + 1 = -\frac{1}{4} = -0.25).

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

(\sin(x)) is increasing on ((0, \frac{\pi}{2})), so (f(x)) is increasing.

Step5: Determine range

Range is ([f(x){\text{min}}, f(x){\text{max}}] = [-\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 ([-0.25, 2.25]).