Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\cot \frac{1}{2} x $$

Step-by-Step Solution

Answer

The period is \(2\pi\), with asymptotes at \(x = 2n\pi\).

Step 1: Identify the Standard Period of Cotangent

The standard period of the cotangent function, \( y = \cot(x) \), is \( \pi \). This period length is due to the regular interval length where the cotangent function repeats its cycle without alteration.

Step 2: Calculate the Period of Transformed Cotangent Function

The given function is \( y = \cot\left(\frac{1}{2}x\right) \). To find the period of this function, divide the standard period by the coefficient of \( x \) inside the function. Therefore, the period is \[ \frac{\pi}{\frac{1}{2}} = 2\pi. \] Hence, the function completes one cycle every \( 2\pi \) units.

Step 3: Determine the Asymptotes of the Function

The asymptotes of \( y = \cot(x) \) occur where the function is undefined, i.e., where \( \sin(x) = 0 \). These are at \( x = n\pi \), where \( n \) is an integer. For \( y = \cot\left(\frac{1}{2}x\right) \), set \( \frac{1}{2}x = n\pi \), so \( x = 2n\pi \). Thus, the asymptotes occur at \( x = 2n\pi \).

Step 4: Sketch the Graph

To sketch the graph: 1. Note that the function passes through zero mid-way between the asymptotes, where \( \cot \left(\frac{1}{2}x \right) \) shifts from \( +\infty \) to \( -\infty \).2. Plot vertical asymptotes at \( x = 2n\pi \), where \( n \) is an integer (e.g., \( x = 0, \pm 2\pi, \pm 4\pi\)).3. The curve decreases from \( +\infty \) to \( -\infty \) in each interval \( 2n\pi < x < (2n+2)\pi \).

Step 5: Verification

Verify by checking one cycle between two asymptotes, e.g., from \( 0 \) to \( 2\pi \), the graph of \( \cot\left( \frac{1}{2}x \right) \) trends downward, intersecting the x-axis at \( \pi \). Confirm this pattern repeats every \( 2\pi \) interval.

Key Concepts

Cotangent Function

The cotangent function, often denoted as \( \cot(x) \), is one of the fundamental trigonometric functions related to a right-angled triangle. It is defined as the reciprocal of the tangent function:

• \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \)
• It is only defined where the sine function \( \sin(x) \) is non-zero.
• Typically seen in the context of a unit circle, \( \cot(x) \) is essentially the ratio of the x-coordinate over the y-coordinate (origin to point on a circle).

The cotangent graph features periodic vertical asymptotes and passes through zeroes at regular intervals. This happens because the sine function, found in the denominator, oscillates between positive and negative values, creating undefined points where it hits zero.
Understanding the simplicity of the cotangent’s rise and fall helps in grasping its plotting and transformations. Like the tangent function, \( \cot(x) \) is periodic and could be compressed or stretched horizontally, depending on the transformations involved.

Function Periodicity

Periodicity is crucial to understanding trigonometric functions like cotangent. The period of a function is the length over which it completes one full cycle before repeating.
For the standard cotangent function \( y = \cot(x) \):

• The period is \( \pi \), due to the repeating nature of sine and cosine functions.
• This means that \( \cot(x) \) repeats its shape every \( \pi \) units across the x-axis.

When a function is transformed, for instance, \( y = \cot(\frac{1}{2}x) \), the period changes based on the coefficient of \( x \). Here, the period calculation involves dividing the original period \( \pi \) by this coefficient:

• Period \( = \frac{\pi}{\frac{1}{2}} = 2\pi \)

The larger period \( 2\pi \) means that the graph of \( \cot \left( \frac{1}{2}x \right) \) takes twice as long to repeat, resulting in horizontal stretching. Recognizing these adjustments helps significantly in plotting and interpreting trigonometric graphs.

Asymptotes

Asymptotes play a significant role in graphing trigonometric functions since they indicate points where the function heads toward infinity or is undefined. For the cotangent function \( y = \cot(x) \),

• Asymptotes occur wherever the function is undefined, specifically where \( \sin(x) = 0 \).
• In the general form, \( x = n\pi \) represents these points, where \( n \) is any integer.

When the function transformation is applied, such as in \( y = \cot\left(\frac{1}{2}x\right) \), it results in different asymptote locations, computed by solving for the transformation equation \( \frac{1}{2}x = n\pi \). Therefore, the asymptotes are now located at:

• \( x = 2n\pi \)

These shifted positions provide the essential vertical dashed lines on the graph, guiding how the function trends between each asymptote, moving from \( +\infty \) to \( -\infty \). Accurately predicting and marking these asymptotes allows for an accurate characterization of the trigonometric function’s behavior.