Problem 4 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\frac{1}{3} \cot x$$

Step-by-Step Solution

Answer

The period of the function is \(\pi\) with vertical asymptotes at \(x = n\pi\).

Step 1: Understanding the Function

The function given is \[ y = \frac{1}{3} \cot x. \] The cotangent function, \(y = \cot x\), is periodic with vertical asymptotes and zeros. We need to find its period and sketch the graph.

Step 2: Identifying the Period

The general form of the cotangent function is \[ y = a \cot(bx + c), \]where the period is given by \( \frac{\pi}{b} \). In our function, \(b = 1\), so the period is \(\pi\). Therefore, the period of \( y = \frac{1}{3} \cot x \) is \(\pi\).

Step 3: Locating Asymptotes

For the cotangent function, vertical asymptotes occur where the function is undefined. Generally, they appear at \[ x = n\pi, \]where \(n\) is an integer. Hence, the vertical asymptotes for our function are at \(x = n\pi\).

Step 4: Amplitude and Vertical Shift

The function \( y = \frac{1}{3} \cot x \) has an amplitude modifier, \( \frac{1}{3} \). This affects the dilation of the graph vertically but does not affect the period or asymptotes. There is no vertical or horizontal shift for this function.

Step 5: Sketching the Graph

To sketch the graph, plot the vertical asymptotes at \( x = n\pi \). Between each pair of asymptotes, draw a cotangent curve that decreases from left to right. The amplitude modifier \( \frac{1}{3} \) will vertically compress the graph compared to the standard \( \cot x \) function.

Key Concepts

Periodicity

Periodicity refers to the repeating nature of trigonometric functions. For the cotangent function, the standard period is \( \pi \). This means that the function's pattern repeats every \( \pi \) units along the x-axis. This property is crucial for predicting the behavior of the function over a long range. Knowing the period helps in sketching the graph, as you only need to plot one interval and replicate it for the others.
It's important to remember the formula for finding the period of a transformed cotangent function. If you have \( y = a \cot(bx + c) \), the period is given by \( \frac{\pi}{b} \). For our function \( y = \frac{1}{3} \cot x \), the value of \( b \) is 1, so the period remains \( \pi \). This consistency affirms that any transformations from the coefficient \( a \) or shifts \( c \) won’t alter the period.

Vertical Asymptotes

Vertical asymptotes are lines where the function heads towards infinity and is undefined, resulting in a gap in the graph. These are critical for the cotangent function, where they regularly occur due to the nature of the function which uses tangents in its definition.
For \( y = \cot x \), vertical asymptotes are found at \( x = n\pi \) where \( n \) is an integer. This is because the cotangent of zero approaches infinity, hence the undefined behavior at these x-values.
In the given function \( y = \frac{1}{3} \cot x \), the asymptotes are still located at \( x = n\pi \). This approach is helpful because it allows us to predict where these undefined points will occur, which is vital when sketching the graph.

Amplitude Modification

Modifying the amplitude of a function changes its height on the y-axis but not its period or phase shift. In trigonometric terms, the amplitude is usually relevant for functions like sine and cosine, where it describes the function’s maximum displacement.
However, for the cotangent function, like \( y = \frac{1}{3} \cot x \), the "amplitude" affects the vertical stretching or compressing of the curve. Here, the number \( \frac{1}{3} \) acts as a vertical compression factor. The cotangent curve will appear less steep, reducing its range vertically between the vertical asymptotes.
Even though the term amplitude isn’t technically used in the same sense as with sine or cosine, the principle of modifier changing the curve’s inclination applies. This slight change influences how the cotangent graph looks but doesn’t affect where it starts repeating or its asymptotic points.

Graphing Trigonometric Functions

Graphing trigonometric functions can initially seem daunting due to various transformations and mathematical behaviors involved. However, breaking it down into simpler steps eases this process.
To sketch \( y = \frac{1}{3} \cot x \), you begin by determining the

• period, which is \( \pi \)
• vertical asymptotes at \( x = n\pi \)

Plot these vertical lines where the function is undefined.
Then draw the cotangent curve between each pair of vertical asymptotes. The curve itself will continually decrease from left to right, depicting the nature of the cotangent function.
The \( \frac{1}{3} \) factor compresses the curve vertically. It’s essential to maintain consistency in the pattern, repeating the visualization over one period. This method helps to grasp spacing and transformation, making the graph easy to understand and accurate.