Problem 25 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The period is \( \frac{\pi}{2} \) with asymptotes at \( x = \frac{(2n-1)\pi}{4} \).

Step 1: Identifying the Basic Function

The given function is related to the cotangent function, which is given by the formula \( y = a \cot(bx + c) \). We recognize the equation \( y = 2 \cot \left(2x + \frac{\pi}{2}\right) \) as a transformation of the standard cotangent function.

Step 2: Determining the Parameter b and the Period

In the standard form of a cotangent function, the period is given by \( \frac{\pi}{b} \). Here, \( b = 2 \), so the period of the function is \( \frac{\pi}{2} \). This is how much the function completes one full cycle or repeats.

Step 3: Finding the Phase Shift

To find the horizontal shift (or phase shift), we use the expression inside the cotangent function, \( bx + c = 0 \). Solving for \( x \) gives \[ 2x + \frac{\pi}{2} = 0 \] which results in \( x = -\frac{\pi}{4} \). This means the graph is shifted left by \( \frac{\pi}{4} \).

Step 4: Identifying the Asymptotes

The vertical asymptotes of \( \cot(x) \) are located at every integer multiple of \( n\pi \). For \( y = 2 \cot(2x + \frac{\pi}{2}) \), the asymptotes occur at the solutions to \( 2x + \frac{\pi}{2} = n\pi \). Solving for \( x \), we find \( x = \frac{n\pi - \frac{\pi}{2}}{2} \).

Step 5: Sketching the Graph

Begin by placing vertical lines at the asymptotes calculated in the previous step. One period of the graph should be sketched between two consecutive asymptotes. The cotangent function has a point of zero (where it intersects the x-axis) halfway between asymptotes.

Key Concepts

Period of a Function

The period of a function tells us how often the function repeats itself along the x-axis. In trigonometric functions, it's the length needed for the function to complete one full cycle. For the cotangent function, expressed generally as \( y = a \cot(bx + c) \), the period can be determined using the formula:

• Period = \( \frac{\pi}{b} \)

In the exercise \( y = 2\cot\left(2x + \frac{\pi}{2}\right) \), we identify that \( b = 2 \). Thus, the period of the function is:

• \( \frac{\pi}{2} \)

This means the cotangent function completes a full cycle every \( \frac{\pi}{2} \) along the x-axis. Understanding the period is crucial for graphing because it shows how frequently the shape of the function repeats itself.

Phase Shift

The phase shift of a trigonometric function represents its horizontal shift along the x-axis. It gives us an idea of where the function's cycle starts compared to its standard position. In the equation \( y = a \cot(bx + c) \), the phase shift is found by setting the argument \( bx + c \) equal to zero:

• \( bx + c = 0 \)
• Solve for \( x \)

For the given function \( y = 2\cot\left(2x + \frac{\pi}{2}\right) \):

• \( 2x + \frac{\pi}{2} = 0 \)
• \( x = -\frac{\pi}{4} \)

This calculation shows that the graph is shifted to the left by \( \frac{\pi}{4} \). By recognizing phase shifts, we can accurately move the starting point of the function along the x-axis when sketching its graph.

Vertical Asymptotes

Vertical asymptotes are lines where the function approaches infinity or negative infinity. They are crucial for the cotangent function where undefined intervals occur due to division by zero. For the function \( y = a \cot(bx + c) \), vertical asymptotes occur at every point where the cotangent function is undefined:

• \( bx + c = n\pi \)

To find the vertical asymptotes for \( y = 2\cot\left(2x + \frac{\pi}{2}\right) \):

• \( 2x + \frac{\pi}{2} = n\pi \)
• Solving for \( x \): \( x = \frac{n\pi - \frac{\pi}{2}}{2} \)

Here, \( n \) is an integer, representing multiple asymptotes spaced within each period. These vertical lines are essential in sketching the graph, outlining where the cotangent function does not exist due to infinite behavior. Recognizing the asymptotes helps in accurately depicting the behavior of the function's graph.