Problem 39 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=-\frac{1}{2} \sec \pi x$$

Step-by-Step Solution

Answer

The given secant function after analyzing will have a period of 2, vertical asymptotes at \(x = \frac{1}{2} + k\) and \(x = \frac{3}{2} + k\), where \(k\) is an integer, and will be reflected about the x-axis due to the negative sign.

Step 1: Identify the Properties of the Function

The given function is \(y = - \frac{1}{2} \sec \pi x\). The secant function is the reciprocal of the cosine function and it is undefined when cosine is equal to zero. Hence, it has vertical asymptotes at these points. Additionally, the secant function has a period of \(2 \pi\). In the given function, \(\pi\) is being used to compress the function horizontally. This means the new period of the function will be \( \frac {2 \pi}{|\pi|} = 2 \). Finally, the negative sign and the \(\frac {1}{2}\) will stretch the function vertically by a factor of -1/2, which will lead to a reflection of the function about the x-axis.

Step 2: Determine the Asymptotes

The cosine function equals zero at \(\frac{\pi}{2}\) and \(\frac{3 \pi}{2}\). Since the period has been compressed by a factor of \(\pi\), we need to adjust these points accordingly. Therefore, these asymptotes when modified due to the horizontal compression become \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\). Since the period is 2, the asymptotes will be \(x = \frac{1}{2} + k\) and \(x = \frac{3}{2} + k\), where \(k\) is an integer.

Step 3: Sketch the Function

For \(x = 0\), calculate the value of \(y\) by substituting \(x = 0\) in the given function which will give \(y = - \frac{1}{2}\). The function will be reflected about the x-axis due to the negative sign. Using these asymptotes and points, sketch the secant function.