the following graph shows a curve that is a transformation of $y = \\sin(x)$. a) what is the period of the…

Explanation:

Step1: Identify period from graph

The period is the horizontal length for one - complete cycle. By observing the graph, the curve repeats itself over an interval of (2\pi). So the period (T = 2\pi).

Step2: Identify amplitude from graph

The amplitude (A) is the distance from the mid - line to the maximum or minimum of the curve. The maximum value is (3) and the minimum value is (- 1). The mid - line is (y = 1). So (A=\frac{3 - (-1)}{2}=2).

Step3: Identify vertical shift from graph

The mid - line of the sine function (y = \sin(x)) is (y = 0). The mid - line of the given curve is (y = 1). So the vertical shift (D = 1).

Step4: Identify phase shift from graph

The sine function (y=\sin(x)) has a zero - crossing at (x = 0). The corresponding zero - crossing of the given curve is at (x=-\frac{\pi}{2}). So the phase shift (C=-\frac{\pi}{2}).

Step5: Write the equation of the curve

The general form of a sinusoidal function is (y = A\sin(B(x - C))+D). Since (B = 1) (because (T=\frac{2\pi}{B}) and (T = 2\pi)), (A = 2), (C=-\frac{\pi}{2}), and (D = 1), the equation is (y=2\sin(x+\frac{\pi}{2})+1).

Answer:

a) (2\pi) b) (2) c) (1) d) (-\frac{\pi}{2}) e) (y = 2\sin(x+\frac{\pi}{2})+1)