which function has the given properties below? the domain is the set of all real numbers. one x - intercept…

which function has the given properties below? the domain is the set of all real numbers. one x - intercept is $(2\\pi,0)$. the amplitude is 4. the point $(\\frac{\\pi}{2},-4)$ is on the graph. the y - intercept is $(0,0)$. $y = -4\\sin(x)$ $y = -4\\cos(x)$ $y = 4\\sin(x)$ $y = 4\\cos(x)$

which function has the given properties below? the domain is the set of all real numbers. one x - intercept is $(2\\pi,0)$. the amplitude is 4. the point $(\\frac{\\pi}{2},-4)$ is on the graph. the y - intercept is $(0,0)$. $y = -4\\sin(x)$ $y = -4\\cos(x)$ $y = 4\\sin(x)$ $y = 4\\cos(x)$

Answer

Explanation:

Step1: Check domain

All the given functions (y = - 4\sin(x)), (y=-4\cos(x)), (y = 4\sin(x)), (y = 4\cos(x)) have a domain of all real - numbers, so this property doesn't help to eliminate any options for now.

Step2: Check x - intercept

For (y=\sin(x)), the x - intercepts are (x = k\pi), (k\in\mathbb{Z}). For (y=\cos(x)), the x - intercepts are (x=\frac{\pi}{2}+k\pi), (k\in\mathbb{Z}). When (x = 2\pi), (y=\sin(2\pi)=0) and (y=\cos(2\pi)=1). So we can eliminate the cosine - based functions (y=-4\cos(x)) and (y = 4\cos(x)) since they do not have an x - intercept at ((2\pi,0)).

Step3: Check amplitude

The general form of a sine or cosine function is (y = A\sin(Bx - C)+D) or (y = A\cos(Bx - C)+D), where (|A|) is the amplitude. For (y=-4\sin(x)) and (y = 4\sin(x)), (|A| = 4), which satisfies the amplitude property.

Step4: Check the point ((\frac{\pi}{2},-4))

For (y = 4\sin(x)), when (x=\frac{\pi}{2}), (y=4\sin(\frac{\pi}{2})=4). For (y=-4\sin(x)), when (x = \frac{\pi}{2}), (y=-4\sin(\frac{\pi}{2})=-4).

Step5: Check y - intercept

For (y=-4\sin(x)), when (x = 0), (y=-4\sin(0)=0). For (y = 4\sin(x)), when (x = 0), (y=4\sin(0)=0).

Answer:

A. (y=-4\sin(x))