Problem 66 - Precalculus Essentials

Question

If you are given the equation of a tangent function, how do you find a pair of consecutive asymptotes?

Step-by-Step Solution

Answer

The tangent function has asymptotes at \((2n + 1) \cdot \frac{\pi}{2}\), where n is an integer. Picking two consecutive integers for n will yield a pair of consecutive asymptotes.

Step 1: Understanding the Periodicity of Tangent Function

The Tangent function is periodic with a period of \( \pi \). This means that the function repeats its values over every interval of \( \pi \). This periodicity is key when working with tangent functions.

Step 2: Finding the Asymptotes

The tangent function has vertical asymptotes at the points \((2n + 1) \cdot \frac{\pi}{2}\) where n is an integer. This implies that for any given value of n, there are vertical asymptotes at these points.

Step 3: Determining Consecutive Asymptotes

To determine a pair of consecutive asymptotes, you would simply choose an integer for n, then calculate the positions of the asymptotes for that integer and for the next integer. For example, if you chose n = 1, the asymptotes would be at \( \frac{3\pi}{2} \) and \( \frac{5\pi}{2} \).