Sketch two periods of the graph for the following function.
j(x) = tan (𝜋/3x)   Identify the asymptotes 



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.

Sketch two periods of the graph for the following function. j(x) = tan(π/3x) Identify the asymptotes.

Learn how to graph two periods of the function j(x) = tan(π/3x) and identify the vertical asymptotes. This solution explains the period of the tangent function and asymptote locations using key trigonometric concepts. Ideal for Grades 10-12 students.

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