Problem 17 - Algebra and Trigonometry

Question

In Exercises 17–24, graph two periods of the given cotangent function. $$ y=2 \cot x $$

Step-by-Step Solution

Answer

The graph of \(y = 2 \cot x\) will have a period of \(\pi\) having vertical asymptotes at \(x = n\pi \) and crossing the x-axis at \(x = n\pi + \frac{\pi}{2} \), repeated twice. It starts from \(\infty\) and decrease to 0 then increase to \(\infty\) in each period.

Step 1: Determine the period and the key points

The standard cotangent function, \(\cot x\), has a period of \(\pi\) (or 180°). This means it repeats its nature after an interval of \(\pi\). The '2' here does not affect the period. For the key points, we know that cotangent is undefined at \(x = n\pi \) where \(n\) is an integer. It is \( \infty \) at these points. Also, at points \( x = n\pi + \frac{\pi}{2} \), cotangent equals to 0.

Step 2: Plot the key points

We plot the points for \(x = 0\), \(x = \frac{\pi}{2}\), \(x = \pi\), \(x = \frac{3\pi}{2}\) and \(x = 2\pi\) for one period. For the second period, we continue on the same pattern i.e. \(x = \frac{5\pi}{2}\), \(x = 3\pi\) etc. At \(x = 0\) and \(x = \pi\), we will have vertical asymptotes, because cotangent approaches \(\infty\) at these points. At \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\), the function will cross the x-axis, because cotangent = 0 at these points.

Step 3: Connect the key points for the graph

Now we can draw our functions connecting these points. The graph decreases from \(\infty\) to 0, crosses the x-axis at \(x = \frac{\pi}{2}\), increases to \(\infty\) until \(x = \pi\), comes from \(-\infty\) at \(x = \pi\), increases to 0 till \(x = \frac{3\pi}{2}\) and then increases to \(\infty\). This forms one period of the cotangent function. For the second period, it will repeat the same pattern.

Key Concepts

Cotangent Function Period

Understanding the period of a cotangent function is crucial when graphing it. The standard cotangent function, denoted as \( \text{cot} \theta \) or \( \text{cot}(x) \), inherently has a period of \( \text{π} \). This means that the function completes its entire set of values within the interval of \( \text{π} \) radians and then repeats its pattern for each subsequent interval of \( \text{π} \). No matter what the coefficient in front of the \( \text{cot}(x) \) function is, in this case '2' as in \( 2 \text{cot} x \), the period remains unchanged.

The practical implication for students graphing the cotangent function is that they should anticipate the function's behavior to repeat every \( \text{π} \) radians along the x-axis, regardless of any vertical stretching or compressing. It's important for students to plot at least one complete period to visualize the function's behavior fully, but including two periods—as the exercise suggests—provides a clearer picture of the function's repeating nature.

Cotangent Function Asymptotes

One of the distinctive features of the cotangent function is the presence of vertical asymptotes. An asymptote is a line that the graph of a function approaches but never actually touches or crosses. For the cotangent function, vertical asymptotes occur because the function is undefined whenever the sine of the angle is zero, meaning at integer multiples of \( \text{π} \), or \( x = n\text{π} \) where \( n \) is an integer. These are precisely the points at which \( \text{cot}(x) \) shoots off towards positive or negative infinity (
\( \text{∞} \) or
\( \text{-∞} \)).

When graphing \( 2 \text{cot} x \), it's essential to clearly mark these asymptotes on the graph since they define the boundaries between which the cotangent function exists. At \( x = 0 \), \( \text{π} \), \( 2\text{π} \), and so on, the function will have vertical lines that the graph will approach but never cross, guiding the overall shape of the cotangent function's graph.

Plotting Trigonometric Functions

When plotting trigonometric functions such as the cotangent, the process involves a combination of identifying characteristic points and understanding the function’s overall behavior. Key points for the cotangent function include where it crosses the x-axis and where it approaches infinity. Cotangent crosses the x-axis at \( x = n\text{π} + \frac{\text{π}}{2} \), where 'n' is an even integer, and it approaches infinity at multiples of \( \text{π} \) radians.

After plotting those key values and asymptotes on a graph, connecting the dots should be done with care to reflect the function's true nature: it decreases from infinity to zero for one half of the period and then increases from negative infinity back to zero for the other half. Instruction of students should emphasize smooth, consistent curves that respect the asymptotes and intercepts. Since cotangent is a periodic function, once you've graphed one complete period, you can replicate this segment for subsequent periods to extend the graph as far as needed.