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Find an equation for f(x) using the secant function. f(x) =

Find an equation for f(x) using the secant function. f(x) =

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Find an equation for \( f(x) \) using the secant function. \( f(x) = \) **Graph Explanation:** The graph displays a periodic function with vertical asymptotes and symmetric properties characteristic of the secant function. It is plotted on a Cartesian coordinate system with the x-axis labeled from \(-2\pi\) to \(2\pi\) and the y-axis labeled from -10 to 10. - **Asymptotes:** Vertical dashed lines indicate asymptotes at \( x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \). - **Behavior:** The graph shows repeating U-shaped branches opening upwards between the asymptotes on the y-axis (for example, from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \)) and inverted U-shaped branches opening downwards below the x-axis (for example, from \( x = \frac{\pi}{2} \) to \( x = \frac{3\pi}{2} \)). - **Periodicity:** The function is periodic with a period of \(2\pi\). - **Amplitude and Phase Shift:** The amplitude of the secant function is not defined due to its vertical asymptotes, and it does not have any horizontal or vertical shifts. This behavior matches the standard form of the secant function, \( f(x) = \sec(x) \), which has vertical asymptotes of this nature and repeats every \( 2\pi \).
Transcribed Image Text:Find an equation for \( f(x) \) using the secant function. \( f(x) = \) **Graph Explanation:** The graph displays a periodic function with vertical asymptotes and symmetric properties characteristic of the secant function. It is plotted on a Cartesian coordinate system with the x-axis labeled from \(-2\pi\) to \(2\pi\) and the y-axis labeled from -10 to 10. - **Asymptotes:** Vertical dashed lines indicate asymptotes at \( x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \). - **Behavior:** The graph shows repeating U-shaped branches opening upwards between the asymptotes on the y-axis (for example, from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \)) and inverted U-shaped branches opening downwards below the x-axis (for example, from \( x = \frac{\pi}{2} \) to \( x = \frac{3\pi}{2} \)). - **Periodicity:** The function is periodic with a period of \(2\pi\). - **Amplitude and Phase Shift:** The amplitude of the secant function is not defined due to its vertical asymptotes, and it does not have any horizontal or vertical shifts. This behavior matches the standard form of the secant function, \( f(x) = \sec(x) \), which has vertical asymptotes of this nature and repeats every \( 2\pi \).
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