The function given is $y = 3 \sec\left(\frac{3}{2}x\right)$, and you are asked to plot points and asymptotes for this secant function.

Step 1: Finding the period and asymptotes

For the general secant function $y = A \sec(Bx)$, the period is given by:

$\text{Period} = \frac{2\pi}{B}$

In this case, $B = \frac{3}{2}$, so the period is:

$\text{Period} = \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3}$

The vertical asymptotes of a secant function occur where the cosine function (the reciprocal of secant) is zero, i.e., where:

$\cos\left(\frac{3}{2}x\right) = 0$

This occurs at:

$\frac{3}{2}x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots$

Solving for $x$, we get the asymptotes at:

$x = \frac{\pi}{3}, x = \pi, x = \frac{5\pi}{3}, \dots$

Step 2: Choosing points to plot

Now that we have the asymptotes, we can choose points between these vertical asymptotes to plot. For example:

1. $x = 0$: $y = 3 \sec\left(\frac{3}{2} \times 0\right) = 3 \sec(0) = 3(1) = 3$

2. $x = \frac{\pi}{6}$ (before the first asymptote at $\frac{\pi}{3}$): $y = 3 \sec\left(\frac{3}{2} \times \frac{\pi}{6}\right) = 3 \sec\left(\frac{\pi}{4}\right) = 3 \times \frac{1}{\cos\left(\frac{\pi}{4}\right)} = 3 \times \sqrt{2} \approx 4.24$

3. $x = \frac{\pi}{2}$ (between the first and second asymptote): $y = 3 \sec\left(\frac{3}{2} \times \frac{\pi}{2}\right) = 3 \sec\left(\frac{3\pi}{4}\right) = 3 \times \frac{1}{-\cos\left(\frac{3\pi}{4}\right)} = -3 \times \sqrt{2} \approx -4.24$

4. $x = \frac{2\pi}{3}$ (just after the second asymptote at $\pi$): $y = 3 \sec\left(\frac{3}{2} \times \frac{2\pi}{3}\right) = 3 \sec(\pi) = 3(-1) = -3$

Step 3: Final Set of Points to Plot

The points you should plot are:

1. $(0, 3)$
2. $\left(\frac{\pi}{6}, 4.24\right)$
3. $\left(\frac{\pi}{2}, -4.24\right)$
4. $\left(\frac{2\pi}{3}, -3\right)$
5. $\left(\frac{5\pi}{6}, 4.24\right)$

These points, combined with the asymptotes at $x = \frac{\pi}{3}, x = \pi$, will help you sketch the graph.

Would you like further details on how to plot this graph or explain more steps?

Questions to Expand Understanding:

1. How does the period of a secant function compare to that of a cosine function?
2. Why do secant functions have vertical asymptotes at certain points?
3. How does the amplitude affect the appearance of the secant graph?
4. Can you determine other asymptotes for this specific function?
5. How would changing the coefficient of $x$ inside the secant function affect the graph?

Tip:

Always check the reciprocal of the secant function (cosine) to easily locate the asymptotes and guide the graph's shape.