Understanding Vertical Asymptotes of Trigonometric Functions

Q5. For the given graphs, which function has a vertical asymptote at a specified value?

Background

Topic: Trigonometric Functions and Asymptotes

This question tests your ability to recognize the graphs of trigonometric functions (such as tangent and cotangent) and identify where their vertical asymptotes occur. Vertical asymptotes are values of $x$ where the function is undefined and the graph approaches infinity.

Key Terms and Formulas

• Vertical Asymptote: A line $x = a$ where the function grows without bound as $x$ approaches $a$.

• Tangent Function: $y = \tan(x)$ has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where $n$ is any integer.

• Cotangent Function: $y = \cot(x)$ has vertical asymptotes at $x = n\pi$, where $n$ is any integer.

• For transformations such as $y = \tan(kx)$ or $y = \cot(kx)$, the period and location of asymptotes change accordingly.

Step-by-Step Guidance

1. Examine each graph and identify the $x$-values where the vertical dashed lines (asymptotes) occur. Note the pattern and spacing between asymptotes.

2. Recall the standard locations of asymptotes for $y = \tan(x)$ and $y = \cot(x)$, and how they shift with transformations (e.g., $y = \tan(kx)$ or $y = \tan(x - a)$).

3. Compare the asymptote locations on each graph to the formulas above. For example, if the asymptotes are at $x = -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}$, consider what transformation of tangent or cotangent would produce these locations.

4. Match the graph with the function whose vertical asymptotes align with the specified value in the question.

Try solving on your own before revealing the answer!