To sketch two periods of the graph for the function $j(x) = \tan\left(\frac{\pi}{3}x\right)$ and identify the vertical asymptotes, we can follow these steps:

1. Understanding the Tangent Function:

The general form of the tangent function is: $y = \tan(kx)$ where $k$ affects the period. The standard period for $\tan(x)$ is $\pi$, but for $j(x) = \tan\left(\frac{\pi}{3}x\right)$, the period will be: $\text{Period} = \frac{\pi}{k} = \frac{\pi}{\frac{\pi}{3}} = 3$ Therefore, the function repeats every 3 units along the x-axis.

2. Vertical Asymptotes:

The vertical asymptotes of the tangent function occur where the function is undefined, which happens when the argument inside the tangent equals an odd multiple of $\frac{\pi}{2}$.

For $j(x) = \tan\left(\frac{\pi}{3}x\right)$, the vertical asymptotes occur when: \frac{\pi}{3}x = \frac{\pi}{2} + n\pi \quad \text{(where n is any integer)} Solving for $x$: $x = \frac{3}{2} + 3n$ Therefore, the vertical asymptotes are located at: $x = \frac{3}{2}, \, x = \frac{9}{2}, \, x = -\frac{3}{2}, \, \text{and so on.}$

3. Sketching Two Periods:

• The period of the function is 3, so the graph will repeat every 3 units.
• Vertical asymptotes will appear at $x = \frac{3}{2}, \frac{9}{2}, -\frac{3}{2}, -\frac{9}{2}$, etc.
• Between the vertical asymptotes, the function will exhibit the typical tangent shape, starting from negative infinity, passing through zero at $x = 0, 3, -3$, and approaching positive infinity as it nears the next asymptote.

Would you like to see a visual sketch of this graph?

5 Related Questions:

1. How does the period of the tangent function change when the coefficient of $x$ changes?
2. What are the key characteristics of a tangent graph?
3. How can vertical asymptotes be found for different trigonometric functions?
4. How does shifting a tangent graph affect its asymptotes?
5. What are other transformations that can be applied to the tangent function?

Tip:

To easily find asymptotes of a tangent function, set the argument of the tangent equal to $\frac{\pi}{2} + n\pi$, as tangent is undefined at these points.