Problem 125 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function. (Include two full periods.) $$f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The graph of the function \( f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right) \) consists of vertical asymptotes at \(x = 0\) and \(x = -\pi\), and it crosses the x-axis at \(x = \frac{\pi}{4}\) and \(x = -\frac{3\pi}{4}\). This pattern repeats every \( \pi \) units and decreases from positive to negative between asymptotes.

Step 1: Identify the properties of the function

Recognize that the function \( f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right) \) is a cotangent function with a vertical shift of 1/2, a horizontal shift of \(-\frac{\pi}{4}\), and no change in period or vertical stretch/shrink factor. The period of a cotangent function is normally \( \pi \). Also note that cotangent function has vertical asymptotes at the end of each period.

Step 2: Determine the critical points

The critical points of the cotangent function are the points at which it crosses the x-axis and vertical asymptotes. Find the x-values for these points. Since the graph is shifted to the left by \(\frac{\pi}{4}\), the x-intercepts and vertical asymptotes will also shift by the same amount. The x-intercepts of the standard cotangent function are \(0\) and \(-\pi\), so in this shifted function, the x-intercepts will be at \(x = \frac{\pi}{4}\) and \(x = -\frac{3\pi}{4}\). Similarly, the asymptotes which were originally at \(x= -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\), will now be at \(x = 0\) and \(x = -\pi\).

Step 3: Sketch the first period

Start the graph at the first vertical asymptote (\(x = 0\)), draw it to the first zero (\(x = \frac{\pi}{4}\)), then draw it to the second vertical asymptote (\(x = -\pi\)). Keep in mind that the cotangent function decreases from positive infinity to negative infinity between asymptotes.

Step 4: Sketch the second period

Repeat the previous step, but this time start at \(x = -\pi\) and end at \(x = -2\pi\). This will complete two full periods of the function.

Key Concepts

Cotangent Function

The cotangent function, represented as \( \ ext{cot}(x) \), is an important trigonometric function closely related to the tangent function. Specifically, it is the reciprocal of the tangent function, meaning it is defined as:

• \( \ ext{cot}(x) = \frac{1}{\text{tan}(x)} = \frac{\text{cos}(x)}{\text{sin}(x)} \)

It is useful in situations where you analyze angles and their ratios, particularly in triangles and periodic phenomena.
For the given function \( f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right) \), the basic cotangent curve is adjusted.
Key adjustments include:

• **Amplitude**: The function is multiplied by \( \frac{1}{2} \), scaling the vertical distances.
• **Horizontal Shift**: The presence of \( +\frac{\pi}{4} \) causes a shift to the left by \( \frac{\pi}{4} \).

Understanding these transformations is crucial when sketching and interpreting the graph of trigonometrical functions.

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals or periods. This property means you only need one cycle to describe the behavior of the entire function.
Trigonometric functions like sine, cosine, and cotangent are naturally periodic.

• The **period** refers to the smallest positive length over which the function's values repeat.
• For cotangent functions, the standard period is \( \pi \), meaning the function repeats every pi units along the x-axis.

In the context of the given function, \( f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right) \), no changes to the period occur.
Thus, two full periods of this function span horizontally from \( x = 0 \) to \( x = -2\pi \).
Graphs of periodic functions help predict future values and are essential in analyzing repetitive phenomena like sound waves and signaling.

Vertical Asymptotes

Vertical asymptotes are lines where a function approaches infinity, causing a break in the graph. They are crucial to identify when graphing rational and trigonometric functions.
For the cotangent function, vertical asymptotes occur where the sine of the angle is zero, i.e., where the tangent function is undefined.

• Typically, these asymptotes appear at \( x = k\pi \), where \( k \) is an integer.

In the modified function \( f(x)=\frac{1}{2} \cot \left(x+\frac{\pi}{4}\right) \), the asymptotes shift due to the horizontal transformation.
They now appear at \( x = 0 \) and \( x = -\pi \) for the first period.
Understanding where these asymptotes lie is essential when plotting, as they indicate where the function does not exist and provide guidance in sketching the curve accurately.