explain how the graph of y = -2sin(x) - 1 differs from the parent function. make sure to compare key points…

Answer:

The graph of (y = - 2\sin(x)-1) differs from the parent - function (y=\sin(x)) in the following ways:

1. Amplitude: The amplitude of the parent function (y = \sin(x)) is (A = 1). For the function (y=-2\sin(x)-1), the amplitude is (|A| = 2). This means that the maximum and minimum values of (y=\sin(x)) are (y = 1) and (y=-1) respectively, while for (y=-2\sin(x)-1), the maximum value is (y=-1 + 2=1) and the minimum value is (y=-1-2=-3).
2. Reflection: The negative sign in front of the (2) in (y=-2\sin(x)-1) reflects the graph of (y = 2\sin(x)) about the (x) - axis. For (y=\sin(x)), when (x = \frac{\pi}{2}), (y = 1); for (y=-2\sin(x)-1), when (x=\frac{\pi}{2}), (y=-2\times1 - 1=-3).
3. Vertical Shift: The (-1) in (y=-2\sin(x)-1) shifts the graph of (y=-2\sin(x)) down by (1) unit. For the parent function (y = \sin(x)), the mid - line is (y = 0). For (y=-2\sin(x)-1), the mid - line is (y=-1).

Some key points for (y=\sin(x)) are:

(x)	(y=\sin(x))
(0)	(0)
(\frac{\pi}{2})	(1)
(\pi)	(0)
(\frac{3\pi}{2})	(-1)
(2\pi)	(0)

For (y=-2\sin(x)-1):

(x)	(y=-2\sin(x)-1)
(0)	(-1)
(\frac{\pi}{2})	(-3)
(\pi)	(-1)
(\frac{3\pi}{2})	(1)
(2\pi)	(-1)

Brief Explanations:

We consider the general form of a sinusoidal function (y = A\sin(Bx - C)+D). Here (A) affects amplitude, the sign of (A) causes reflection, and (D) is for vertical shift. We compare key - points of the parent function (y=\sin(x)) and the transformed function (y=-2\sin(x)-1) to show the differences.