Problem 50 - Precalculus Essentials

Question

Graph two periods of each function. $$y=\sec |x|$$

Step-by-Step Solution

Answer

The secant of the absolute value of x is a periodic wave that is mirrored across the y-axis. It has vertical asymptotes and two branches per period like the conventional secant function, but due to the absolute value of x, there is symmetry about the y-axis as well.

Step 1: Understand the Secant Function

Firstly, have a look at the secant function. It's \(y=\sec x\), which can be written as \(y=1/\cos x\). The secant function has vertical asymptotes and two branches per period between \(x=-\pi/2\) and \(x=\pi/2\).

Step 2: Understand absolute value

The absolute value of any input is always non-negative. In other words, this means if \(x\) is negative, we change it to positive; otherwise, we keep \(x\) as is. The absolute value function reflects the graphic on the negative side of the x-axis to the positive side, which means it mirrors everything over the y-axis when we deal with \(|x|\).

Step 3: Graph the function

Given the shape and properties of the secant function and notes on the absolute value in previous steps, we can graph the function \(y=\sec |x|\). It's just like \(y=sec(x)\), but mirrored on the y-axis for negative \(x\). To graph, draw two branches per period between \(x=-\pi/2\) and \(x=\pi/2\), and then repeat these branches for \(x=-3\pi/2\) to \(x=-\pi/2\) and \(x=\pi/2\) to \(x=3\pi/2\). Remember to mark the asymptotes and the x-intercepts.