Question

5. Graph the function f (x) = 3sin (2x) + 1 Be sure to identify the midline, period, and amplitude.

Answer 1

Given that f(x) = 3 sin (2x) + 1

Given that : a sin (bx + c ) + d

let a = amplitude,

Midline is the that runs between the maximum and minimum value

`\begin{gathered} \text{ Since, amplitude = 3} \\ \text{the graph is shifted 1 unit in positive y - coordinate} \\ \text{Maximum value = 3 - 1 = 2} \\ \text{ minimum value = -3 - 1 = -4} \\ \text{Midline is the center of (2, - 4)} \\ \text{Midline = }\frac{\text{2 - 4}}{2} \\ \text{midline = -1} \end{gathered}`

Period is calculated as

`\begin{gathered} \text{period = }(2\pi)/(|b|) \\ \\ \text{b = 2} \\ \text{Period = }(2\pi)/(2) \\ \text{Period = }\pi\text{second} \end{gathered}`

Frequency = 1 / period

`\text{frequency = }(1)/(\pi)\text{ Hz}`