graph the following function\n$y = 3cos(x+\frac{pi}{10})$ \nuse the graphing tool to graph the…

Explanation:

Step1: Identify the general form

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. For the function $y = 3\cos(x+\frac{\pi}{10})$, we have $A = 3$, $B = 1$, $C=-\frac{\pi}{10}$, and $D = 0$.

Step2: Find the amplitude

The amplitude $|A|$ gives the maximum displacement from the mid - line. Here, $|A|=|3| = 3$.

Step3: Find the period

The period of the cosine function $y = A\cos(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Since $B = 1$, the period $T=\frac{2\pi}{|1|}=2\pi$.

Step4: Find the phase shift

The phase shift is given by $\frac{C}{B}$. Here, $\frac{C}{B}=\frac{-\frac{\pi}{10}}{1}=-\frac{\pi}{10}$, which means the graph of $y = 3\cos(x)$ is shifted to the left by $\frac{\pi}{10}$ units.

Step5: Plot key points

We know that for the basic cosine function $y=\cos(x)$, key points in one period $[0, 2\pi]$ are $(0,1),(\frac{\pi}{2},0),(\pi,- 1),(\frac{3\pi}{2},0),(2\pi,1)$. For the function $y = 3\cos(x+\frac{\pi}{10})$, we adjust the $x$ - values according to the phase shift.

• When $x+\frac{\pi}{10}=0$, $x =-\frac{\pi}{10}$, and $y = 3$.
• When $x+\frac{\pi}{10}=\frac{\pi}{2}$, $x=\frac{\pi}{2}-\frac{\pi}{10}=\frac{5\pi - \pi}{10}=\frac{2\pi}{5}$, and $y = 0$.
• When $x+\frac{\pi}{10}=\pi$, $x=\pi-\frac{\pi}{10}=\frac{9\pi}{10}$, and $y=-3$.
• When $x+\frac{\pi}{10}=\frac{3\pi}{2}$, $x=\frac{3\pi}{2}-\frac{\pi}{10}=\frac{15\pi-\pi}{10}=\frac{7\pi}{5}$, and $y = 0$.
• When $x+\frac{\pi}{10}=2\pi$, $x=2\pi-\frac{\pi}{10}=\frac{19\pi}{10}$, and $y = 3$.

Then, use a graphing tool to plot these points and draw a smooth cosine - curve with amplitude 3, period $2\pi$, and phase shift $-\frac{\pi}{10}$.

Since the problem mainly asks to use a graphing tool, the above steps are for understanding the function's characteristics before graphing. When using the graphing tool, input the function $y = 3\cos(x+\frac{\pi}{10})$ directly.

Answer:

Use a graphing tool to input the function $y = 3\cos(x+\frac{\pi}{10})$ and graph it according to the key - point analysis above.