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

Answer:

B. $y = - 3\cos(x)$

Explanation:

Step1: Check domain

All given functions $y = - 3\sin(x)$, $y=-3\cos(x)$, $y = 3\sin(x)$, $y = 3\cos(x)$ have domain of all real - numbers.

Step2: Check x - intercept

For $y = A\sin(x)$ or $y = A\cos(x)$, the x - intercepts of $y=\sin(x)$ are $x = k\pi,k\in\mathbb{Z}$ and of $y=\cos(x)$ are $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$. For $y=-3\cos(x)$, when $x = \frac{\pi}{2}$, $y=-3\cos(\frac{\pi}{2})=0$.

Step3: Check maximum value

The general form of a sine or cosine function is $y = A\sin(x)$ or $y = A\cos(x)$ with amplitude $|A|$. The maximum value of $y = A\sin(x)$ and $y = A\cos(x)$ is $|A|$. For $y=-3\cos(x)$, $|A| = 3$, so the maximum value is 3.

Step4: Check y - intercept

For the y - intercept, we set $x = 0$. For $y=-3\cos(x)$, when $x = 0$, $y=-3\cos(0)=-3$.