AP Precalculus Practice Test: Unit 3 Question #38 Find the Vertical Asymptotes of f(x) = 2sec(x) - 5

Please subscribe!    / nickperich   My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May. *AP Precalculus Practice Test: Unit 3, Question #38* asks you to find the vertical asymptotes of the function \( f(x) = 2\sec(x) - 5 \). --- *Key Concepts* 1. **Secant Function**: The secant function, \( \sec(x) \), is the reciprocal of the cosine function: \[ \sec(x) = \frac{1}{\cos(x)} \] The secant function has vertical asymptotes wherever \( \cos(x) = 0 \) because the reciprocal of zero is undefined. 2. **Finding Vertical Asymptotes**: To find the vertical asymptotes of \( f(x) = 2\sec(x) - 5 \), we need to determine where \( \sec(x) \) (or equivalently \( \cos(x) \)) is undefined, i.e., where \( \cos(x) = 0 \). --- *Solving the Problem* 1. **Set the cosine function equal to 0**: \( \sec(x) \) is undefined wherever \( \cos(x) = 0 \). The values of \( x \) where \( \cos(x) = 0 \) occur at: \[ x = \frac{\pi}{2} + n\pi, \quad \text{where} \quad n \in \mathbb{Z} \] This is because cosine is 0 at odd multiples of \( \frac{\pi}{2} \). 2. **Find the vertical asymptotes**: The vertical asymptotes of \( f(x) = 2\sec(x) - 5 \) occur at the same values of \( x \) where \( \cos(x) = 0 \), i.e., the values where \( x = \frac{\pi}{2} + n\pi \). --- **Summary**: The vertical asymptotes of \( f(x) = 2\sec(x) - 5 \) occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. These are the points where \( \cos(x) = 0 \), causing the secant function to be undefined. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study