Problem 24 - Precalculus Essentials

Question

Graph two periods of the given cotangent function. $$y=3 \cot \left(x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The graph of the function \( y = 3 \cot (x + \pi/4) \) has vertical asymptotes at \( x = -\pi/4 \) and \( x = 3\pi/4 \), forms the shape of a decreasing function from \(-\pi/4\) to \(3\pi/4\) in the interval and continues the same pattern for every \( \pi \) interval on the x-axis.

Step 1: Find the amplitude, period, and phase shift

In this exercise, we have the function \( y = 3 \cot (x + \pi/4) \). The number multiplying \( \cot \) is the amplitude and in our function, the amplitude is 3. The value inside the parenthesis \( x + \pi/4 \) represents a phase-shift of \(-\pi/4\) to the left. Note, a positive value would shift the function to the right. The period of cotangent function is \( \pi \), the function repeats after an interval of \( \pi \). Here, there is no horizontal stretch or compression, hence, the period stays the same as \( \pi \).

Step 2: Draw the basic shape of the cotangent function

The cotangent function has vertical asymptotes at multiples of \( \pi \) because cotangent is undefined at those values. It's always good to start by drawing these asymptotes. The graph normally starts at the point (\( 0, 0 \)) but due to the shift, our graph will start at \(-\pi/4\). Cotangent decreases from \( \infty \) down to \( -\infty \) as \( x \) increases.

Step 3: Plot points and sketch the graph

Plot some key points for one period of the function. Start at x = \(-\pi/4\) since we have a shift of \(-\pi/4\). Note that at x = \(-\pi/4\) function approaches infinity due to asymptote. Midway i.e at \( \pi/2 - \pi/4 = \pi/4 \), function will be 0 and at \( \pi - \pi/4 = 3\pi/4 \), function will again approach negative infinity. Now, sketch the curve that fits these points for one period from \(-\pi/4\) to \( 3\pi/4 \). Repeat the shape of this curve to the right and to the left as many times as necessary to draw 2 periods of the function. The curve will appear to hug the asymptotes but never touch them.