📚 Topic Summary
The inverse tangent function, denoted as $arctan(x)$ or $tan^{-1}(x)$, returns the angle whose tangent is $x$. Its graph is a transformation of the standard tangent function, reflected and restricted to the range $(-\frac{\pi}{2}, \frac{\pi}{2})$. Understanding the key features such as asymptotes and intercepts is crucial for accurately graphing inverse tangent functions. Transformations like shifts and stretches also play a significant role. Remember, the domain of $arctan(x)$ is all real numbers, and its range is between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ (exclusive).
Graphing these functions often involves identifying key points and asymptotes. Shifts and stretches will affect the location of these features, so careful consideration is needed to ensure the graph is accurate. This worksheet will walk you through the vocabulary, the concepts and give you a chance to apply what you learn!
🧮 Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Inverse Tangent |
A. The vertical lines that the graph approaches but never touches. |
| 2. Asymptote |
B. The value of x where the graph crosses the x-axis. |
| 3. X-intercept |
C. The function that returns the angle whose tangent is a given number. |
| 4. Range |
D. The set of all possible output values of a function. |
| 5. Domain |
E. The set of all possible input values of a function. |
✏️ Part B: Fill in the Blanks
The inverse tangent function, also known as ___________, has a range of ___________, and its domain is ___________. The graph has horizontal asymptotes at $y = -\frac{\pi}{2}$ and $y = \frac{\pi}{2}$. Transformations to the function, such as vertical stretches or horizontal shifts, will affect the shape and position of the ___________. For example, the function $f(x) = 2 \cdot arctan(x)$ has the same __________ as arctan(x) but different __________.
🤔 Part C: Critical Thinking
Explain how understanding the transformations of the standard tangent function can help you quickly sketch the graph of a transformed inverse tangent function. Provide an example to illustrate your explanation.