Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\cot \left(x-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Period: \(\pi\). Asymptotes at \(x = k\pi + \frac{\pi}{2}\).

Step 1: Identify the Function and its Form

The given function is a cotangent function: \(y = \cot\left(x - \frac{\pi}{2}\right)\). This is a horizontally shifted form of \(y = \cot x\). The shift is \(\frac{\pi}{2}\) units to the right.

Step 2: Determine the Base Period

The base period of \(y = \cot x\) is \(\pi\). This means that one complete cycle of the cotangent function occurs every \(\pi\) units along the x-axis.

Step 3: Analyze the Phase Shift

The term \(x - \frac{\pi}{2}\) represents a phase shift. Specifically, since it's in the form \(x - C\), it means the graph shifts horizontally by \(\frac{\pi}{2}\) units to the right.

Step 4: Identify the Asymptotes of \(y = \cot x\)

For \(y = \cot x\), the vertical asymptotes occur where the function is undefined, at \(x = k\pi\), where \(k\) is an integer. This means asymptotes appear at integer multiples of \(\pi\).

Step 5: Adjust Asymptotes for the Shift

Applying the horizontal shift of \(\frac{\pi}{2}\), the new asymptotes of \(y = \cot(x - \frac{\pi}{2})\) occur at \(x = k\pi + \frac{\pi}{2}\).

Step 6: Sketch the Graph

To sketch \(y = \cot(x - \frac{\pi}{2})\), start by plotting the asymptotes at each \(x = k\pi + \frac{\pi}{2}\). The cotangent curve will rise from negative infinity to positive infinity between consecutive asymptotes, crossing the x-axis halfway between them.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are widely used to analyze periodic phenomena.
The cotangent, denoted as \( \cot(x) \), is one of these functions. It is closely related to the sine and cosine functions, being the reciprocal of the tangent function.
The basic properties of the cotangent function include:

• It is undefined when the tangent is zero, which leads to vertical asymptotes.
• Its graph oscillates between positive and negative infinity without upper or lower bounds.
• Like other trigonometric functions, it repeats at regular intervals, making it periodic.

Understanding these properties helps in sketching and analyzing transformations of the cotangent function.

Graphing Transformations

Graphing transformations involve shifting, stretching, or reflecting the graph of a function.
For the cotangent function \( y = \cot(x) \), transformations can include horizontal shifts, such as seen in the function \( y = \cot(x - \frac{\pi}{2}) \).
This specific equation represents a horizontal shift by \( \frac{\pi}{2} \) units to the right.
When graphing transformations:

• Identify shifts in the x-direction or y-direction.
• Adjust the graph accordingly by shifting every point of the base function.
• Maintain the shape and relative positions of critical features like asymptotes and intercepts.

Being thorough with these transformations allows for accurate representation of the function's graph.

Periodicity

Periodicity is an essential feature of trigonometric functions like cotangent.
The cotangent function has a base period of \( \pi \), meaning it completes one full cycle over this interval.
The concept of periodicity involves:

• The function value repeating at regular intervals.
• Graphical features such as peaks, troughs, and asymptotes occurring consistently.

Even when transformations occur, such as horizontal shifts, the underlying periodicity remains unchanged.
For instance, \( y = \cot(x - \frac{\pi}{2}) \) still has a period of \( \pi \) despite the shift.
Recognizing this trait is crucial for predicting the behavior of trigonometric graphs.

Vertical Asymptotes

Vertical asymptotes in functions occur where the output tends to infinity, typically due to dividing by zero.
For the cotangent function, vertical asymptotes mark the points where the function is undefined.
In \( y = \cot x \), these asymptotes occur at \( x = k\pi \) (where \( k \) is an integer).
Shifts in the function, such as \( y = \cot(x - \frac{\pi}{2}) \), alter the location of these asymptotes.
Following the horizontal shift, the new asymptotes are found at \( x = k\pi + \frac{\pi}{2} \).
Key points to remember:

• Vertical asymptotes occur where the function's denominator equals zero.
• They guide the placement of the function's graph.
• Transformations like shifts move the asymptotes accordingly.

These asymptotes are critical for sketching accurate graphs of the transformed function.