Question

graph the trigonometric function. y = 3 sin(\frac{3}{4}x)-1. plot all points corresponding to minima and maxima within one cycle. within that cycle, also plot all points along the \midline\ (points whose y - coordinates are midway between the functions minimum and maximum values). then click on the graph - a - function button.

Explanation:

Step1: Recall properties of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $y = 3\sin(\frac{3}{4}x)-1$, we have $A = 3$, $B=\frac{3}{4}$, $C = 0$, and $D=-1$.

Step2: Find the amplitude

The amplitude is given by $|A|$. Here, $|A| = 3$. The maximum value of $y = 3\sin(\frac{3}{4}x)-1$ is $y_{max}=3 - 1=2$ and the minimum value is $y_{min}=-3 - 1=-4$.

Step3: Find the mid - line

The mid - line of the function $y = A\sin(Bx - C)+D$ is $y = D$. So, the mid - line of $y = 3\sin(\frac{3}{4}x)-1$ is $y=-1$.

Step4: Find the period

The period of the sine function $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Since $B=\frac{3}{4}$, then $T=\frac{2\pi}{\frac{3}{4}}=\frac{8\pi}{3}$.

Step5: Find key points

• Maxima: $\sin(\frac{3}{4}x)=1$, so $\frac{3}{4}x=\frac{\pi}{2}+2k\pi,k\in\mathbb{Z}$. Solving for $x$, we get $x=\frac{2\pi}{3}+\frac{8k\pi}{3},k\in\mathbb{Z}$. In one period ($k = 0$), when $x=\frac{2\pi}{3}$, $y = 2$.
• Minima: $\sin(\frac{3}{4}x)=-1$, so $\frac{3}{4}x=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}$. Solving for $x$, we get $x = 2\pi+\frac{8k\pi}{3},k\in\mathbb{Z}$. In one period ($k = 0$), when $x = 2\pi$, $y=-4$.
• Mid - line points: When $\frac{3}{4}x = k\pi,k\in\mathbb{Z}$, $x=\frac{4k\pi}{3},k\in\mathbb{Z}$. In one period, for example, when $x = 0$, $y=-1$; when $x=\frac{4\pi}{3}$, $y=-1$; when $x=\frac{8\pi}{3}$, $y=-1$.

To graph the function, plot the points: maxima $(\frac{2\pi}{3},2)$, minima $(2\pi,-4)$ and mid - line points $(0, - 1),(\frac{4\pi}{3},-1),(\frac{8\pi}{3},-1)$ within one period $[0,\frac{8\pi}{3}]$ and then repeat the pattern for other periods.

Answer:

Plot the points $(\frac{2\pi}{3},2),(2\pi,-4),(0, - 1),(\frac{4\pi}{3},-1),(\frac{8\pi}{3},-1)$ and repeat the pattern for other periods.