To sketch the graph of the function $p(x) = \tan(x - \frac{\pi}{4})$, let's break down the problem step by step:

1. General Properties of the Tangent Function:

The general tangent function is $\tan(x)$. For this function, the important properties are:

• Period: The period of $\tan(x)$ is $\pi$, meaning that the graph repeats every $\pi$ units along the x-axis.
• Asymptotes: The asymptotes of $\tan(x)$ occur where the function is undefined, i.e., at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

2. Effect of the Transformation $x - \frac{\pi}{4}$:

The transformation $p(x) = \tan(x - \frac{\pi}{4})$ represents a horizontal shift of the graph of $\tan(x)$ by $\frac{\pi}{4}$ units to the right. This means:

• The asymptotes will shift from $x = \frac{\pi}{2} + n\pi$ to $x = \frac{\pi}{4} + n\pi$.
• The period remains $\pi$ because the coefficient of $x$ inside the tangent function is 1 (no horizontal stretching).

3. Key Features:

• Period: The period of the graph is still $\pi$, as the transformation only involves a shift, not a stretching or compressing.
• Asymptotes: The vertical asymptotes of the function will occur at: $x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}$ So, the first two asymptotes are at $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$.

4. Sketching Two Periods:

• For the first period (from $x = \frac{\pi}{4}$ to $x = \frac{5\pi}{4}$):

  • The tangent function will cross through $x = \frac{\pi}{2}$, the midpoint between the asymptotes.
  • The graph will start from the asymptote at $x = \frac{\pi}{4}$, rise to the midpoint, and then approach the asymptote at $x = \frac{5\pi}{4}$.

• For the second period (from $x = \frac{5\pi}{4}$ to $x = \frac{9\pi}{4}$):

  • The behavior is similar: the graph rises from $x = \frac{5\pi}{4}$, reaches the midpoint at $x = \frac{7\pi}{4}$, and approaches the asymptote at $x = \frac{9\pi}{4}$.

5. Final Graph:

You will see a standard tangent shape, but shifted to the right by $\frac{\pi}{4}$.

Summary of Key Features:

• Period: $\pi$ (unchanged).
• Asymptotes: Occur at $x = \frac{\pi}{4} + n\pi$, where $n$ is an integer.
• Stretching Factor: There is no stretching factor, as the coefficient of $x$ is 1.

Would you like a visual representation or any further explanation on the behavior of this function?

Related Questions:

1. How does a horizontal shift affect the tangent function?
2. What happens to the period of a tangent function if there's a coefficient inside the argument, like $\tan(kx)$?
3. How do you find the asymptotes of the function $\tan(x + c)$?
4. What happens if the function is stretched vertically (e.g., $a \tan(x)$)?
5. Can you graph other trigonometric functions, like sine and cosine, with similar transformations?

Tip: To graph any transformation of $\tan(x)$, always start by identifying the asymptotes, then plot the midpoints where the function crosses the x-axis, and finally sketch the standard shape of the tangent curve between them.