How to graph y = tan x step-by-step for Pre-Calculus

📚 Understanding the Tangent Function

The tangent function, denoted as $y = \tan x$, is a fundamental trigonometric function. It's defined as the ratio of the sine function to the cosine function: $\tan x = \frac{\sin x}{\cos x}$. Graphing the tangent function requires understanding its periodic nature, asymptotes, and key points.

📜 History and Background

The tangent function has been used for centuries in trigonometry and surveying. Its origins can be traced back to the study of triangles and angles in ancient Greece. The modern notation and understanding of the tangent function were developed during the Middle Ages and the Renaissance, with contributions from mathematicians like Brahmagupta and later European scholars.

🔑 Key Principles for Graphing $y = \tan x$

• 📐 Periodicity: The tangent function has a period of $\pi$. This means that the graph repeats itself every $\pi$ units along the x-axis. Mathematically, $\tan(x + \pi) = \tan(x)$.
• 🧱 Asymptotes: The tangent function has vertical asymptotes where $\cos x = 0$. This occurs at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer. These asymptotes are crucial for defining the shape of the graph.
• 📍 Key Points: It's helpful to identify key points such as where $\tan x = 0$, which occurs at $x = n\pi$, where $n$ is an integer. Also, consider points where $\tan x = 1$ (e.g., $x = \frac{\pi}{4}$) and $\tan x = -1$ (e.g., $x = -\frac{\pi}{4}$).
• 📈 Behavior: As $x$ approaches an asymptote from the left, $\tan x$ approaches positive infinity. As $x$ approaches an asymptote from the right, $\tan x$ approaches negative infinity.

✍️ Step-by-Step Guide to Graphing $y = \tan x$

1. 🧱 Identify Asymptotes:
   • 🧭 Find the values of $x$ where $\cos x = 0$. These are the vertical asymptotes. For example, $x = \frac{\pi}{2}$ and $x = -\frac{\pi}{2}$ are two common asymptotes.
2. 📍 Find Key Points:
   • 🧮 Determine where $\tan x = 0$. This occurs at integer multiples of $\pi$ (i.e., $x = n\pi$). For instance, $x = 0$, $x = \pi$, $x = -\pi$.
   • ➕ Find points where $\tan x = 1$. This occurs at $x = \frac{\pi}{4} + n\pi$.
   • ➖ Find points where $\tan x = -1$. This occurs at $x = -\frac{\pi}{4} + n\pi$.
3. 📈 Sketch the Graph:
   • ✏️ Draw the vertical asymptotes as dashed lines.
   • 📍 Plot the key points you identified.
   • 〰️ Sketch the curve, ensuring it approaches positive infinity as it nears an asymptote from the left and negative infinity as it nears an asymptote from the right.
   • 🔁 Repeat the pattern for each period of $\pi$.

🌍 Real-World Examples

• 📡 Navigation: The tangent function is used in navigation to calculate angles and distances, especially in surveying and mapping.
• 💡 Engineering: It appears in various engineering applications, such as determining the slope of a line or analyzing the behavior of oscillating systems.
• 👓 Optics: The tangent function is used in optics to describe the angle of refraction of light as it passes through different mediums.

🧪 Practice Problems

Here are some practice problems to test your understanding:

1. ❓ Graph $y = \tan(x - \frac{\pi}{4})$.
2. ❓ Graph $y = 2\tan x$.
3. ❓ Graph $y = -\tan x$.

📝 Conclusion

Graphing $y = \tan x$ involves understanding its periodicity, asymptotes, and key points. By following these steps, you can accurately sketch the graph and apply this knowledge to real-world applications. Remember to practice regularly to reinforce your understanding!