Question

graph the trigonometric function. y = 3 sin 2x+1. plot all points corresponding to minima and maxima within one cycle. within that cycle, also plot all points along the \mid - line\ (points whose y - coordinates are midway between the functions minimum and maximum values). then click on the graph - a - function button.

Explanation:

Step1: Recall the general form of a sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $y = 3\sin(2x)+1$, we have $A = 3$, $B = 2$, $C = 0$, and $D = 1$.

Step2: Find the amplitude, period, mid - line, maximum, and minimum values

The amplitude $|A|=3$. The period $T=\frac{2\pi}{B}=\frac{2\pi}{2}=\pi$. The mid - line is given by $y = D$, so $y = 1$. The maximum value of $y$ is $y_{max}=A + D=3 + 1=4$ and the minimum value is $y_{min}=-A + D=-3 + 1=-2$.

Step3: Find key points within one period

For $y = 3\sin(2x)+1$, when $2x = 0$, $x = 0$ and $y=3\sin(0)+1 = 1$. When $2x=\frac{\pi}{2}$, $x=\frac{\pi}{4}$ and $y = 3\sin(\frac{\pi}{2})+1=4$. When $2x=\pi$, $x=\frac{\pi}{2}$ and $y = 3\sin(\pi)+1 = 1$. When $2x=\frac{3\pi}{2}$, $x=\frac{3\pi}{4}$ and $y=3\sin(\frac{3\pi}{2})+1=-2$. When $2x = 2\pi$, $x=\pi$ and $y=3\sin(2\pi)+1 = 1$.

Step4: Plot the points

• Maxima: The maxima occur when $\sin(2x)=1$, i.e., $2x=\frac{\pi}{2}+2k\pi,k\in\mathbb{Z}$, so $x=\frac{\pi}{4}+k\pi$. For one - cycle ($0\leq x\leq\pi$), the maximum point is $(\frac{\pi}{4},4)$.
• Minima: The minima occur when $\sin(2x)=-1$, i.e., $2x=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}$, so $x=\frac{3\pi}{4}+k\pi$. For one - cycle ($0\leq x\leq\pi$), the minimum point is $(\frac{3\pi}{4},-2)$.
• Mid - line points: When $2x = 0,\pi,2\pi$ (i.e., $x = 0,\frac{\pi}{2},\pi$), $y = 1$. So the mid - line points in the interval $[0,\pi]$ are $(0,1),(\frac{\pi}{2},1),(\pi,1)$.

Answer:

Plot the points $(0,1),(\frac{\pi}{4},4),(\frac{\pi}{2},1),(\frac{3\pi}{4},-2),(\pi,1)$ on the graph and connect them with a smooth curve to represent one - cycle of the function $y = 3\sin(2x)+1$. The mid - line is the horizontal line $y = 1$, the maximum value in one - cycle is $y = 4$ at $x=\frac{\pi}{4}$ and the minimum value is $y=-2$ at $x=\frac{3\pi}{4}$.