Problem 34 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=3 \sec x$$

Step-by-Step Solution

Answer

The graph of the function \(y=3 sec x\) has two periods from \(-2\pi\) to \(2\pi\) with an amplitude of 3. The graph approaches asymptotes where \(x = k\pi/2\), where \(k\) is an odd integer, and repeats the pattern every \(2\pi\).

Step 1: Recognize the Parent Function and Amplitude

The parent function is \(sec(x)\) which is the reciprocal of \(cos(x)\). The given function \(y = 3sec(x)\) has an amplitude of 3, which scales the secant function vertically by a factor of 3.

Step 2: Identify the Key Points.

The key points for the secant function are the maximum, minimum, and undefined points that occur at \( x = k\pi/2 \), where \(k\) is an odd integer. We then need to multiply the y-values of these key points by the amplitude of 3 in the given function.

Step 3: Sketch the Function.

First graph the cosine function like a dashed line between \(-2\pi\) and \(2\pi\), where \(x = -3\pi/2, -\pi/2, \pi/2, 3\pi/2\) are the zeroes of the function. Then plot the key points calculated in Step 2. Connect the points with a continuous curve that approaches asymptotes where the functioning is undefined. Repeat the process for a second period of \(2\pi\) until \(x = 2\pi\). This graph represents two periods of the given secant function \(y = 3sec(x)\).