Question

question 7 (5 points) listen which of the following are maximum and minimum points of the function y = sin(x + π)? a) maximum: (π/2, 1); minimum: (3π/2, -1) b) minimum: (π/2, - 1); maximum: (3π/2, 1) c) minimum: (π/4, 1); maximum: (5π/4, -1) d) minimum: (π/4, - 1); maximum: (5π/4, 1) question 8 (5 points) listen b 16 a 28

Explanation:

Step1: Recall sine - function properties

The general form of a sine function is $y = A\sin(Bx - C)+D$. For the function $y=\sin(x + \pi)$, we know that the maximum value of the sine - function $y = \sin(u)$ is 1 and the minimum value is - 1.
We know that $\sin(u)=1$ when $u=\frac{\pi}{2}+2k\pi,k\in\mathbb{Z}$ and $\sin(u)= - 1$ when $u=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}$.

Step2: Solve for $x$ when $y$ is maximum

Set $x+\pi=\frac{\pi}{2}+2k\pi$. Solving for $x$ gives $x=\frac{\pi}{2}-\pi+2k\pi=-\frac{\pi}{2}+2k\pi$. When $k = 1$, $x=\frac{3\pi}{2}$, and $y = 1$.

Step3: Solve for $x$ when $y$ is minimum

Set $x+\pi=\frac{3\pi}{2}+2k\pi$. Solving for $x$ gives $x=\frac{3\pi}{2}-\pi+2k\pi=\frac{\pi}{2}+2k\pi$. When $k = 0$, $x=\frac{\pi}{2}$, and $y=-1$.

Answer:

B. Minimum: $(\frac{\pi}{2}, - 1)$; Maximum: $(\frac{3\pi}{2},1)$