Problem 22 - Algebra and Trigonometry

Question

In Exercises 17–24, graph two periods of the given cotangent function. $$ y=-2 \cot \frac{\pi}{4} x $$

Step-by-Step Solution

Answer

The graphing properties of this cotangent function are: amplitude of 2, period of 4, frequency of 1/4, and it is a reflection over the x-axis. The graph starts at a vertical asymptote, then passes through the middle of the period at the x-axis, and finally ends at another vertical asymptote.

Step 1: Identify the Period and Frequency

The period of the cotangent function in the equation \(y = -2 cot(\frac{\pi}{4}x)\) can be identified by taking the reciprocal of the coefficient of x inside the cotangent function. Thus, the period is \(P = 4\) and frequency is \(\frac{1}{4}\).

Step 2: Identify the Amplitude

The amplitude of the cotangent function is the absolute value of the coefficient of the cotangent function. For the given function, the amplitude is \( |-2| = 2 \).

Step 3: Sketch the Period

The period of the cotangent function starts and ends with vertical asymptotes. For our function, one period is from 0 to 4. Therefore, the asymptotes of this function are every multiple of 4 or at \(x=4n\) where n is an integer.

Step 4: Sketch the Graph

The cotangent graph starts from the top at zero and goes down touching x-axis exactly at the middle of the period. For this function it will touch x-axis at 2. As the cotangent function is negative here, the graph will be the reflection of cotangent graph about x-axis.

Key Concepts

Trigonometric Functions

Trigonometric functions form the foundation of trigonometry, a branch of mathematics that deals with the relationships between angles and sides of triangles. Among these functions are sine (sin), cosine (cos), tangent (tan), along with their reciprocals: cosecant (csc), secant (sec), and cotangent (cot).

These functions are crucial for modeling periodic phenomena such as sound waves, light waves, and the motion of pendulums. Specifically, the cotangent function, which is the reciprocal of the tangent function, shows the ratio of the adjacent side to the opposite side of a right-angled triangle. In the case of graphing cotangent functions, we depict the variation of this ratio as the angle changes, which results in a repeating wave-like pattern.

Cotangent Period and Frequency

The period of a trigonometric function is the length of one complete cycle of the wave, after which the pattern repeats. For the cotangent function, the period is determined by the reciprocal of the coefficient found with the variable within the function. In the example of the given function, \( y = -2 \cot(\frac{\pi}{4}x) \), the coefficient of x inside the cotangent is \( \frac{\pi}{4} \) which gives us a period \( P = 4 \).

Frequency is the number of cycles the function completes in a unit interval and is the reciprocal of the period. So, if the period is 4, the frequency is \( \frac{1}{4} \) cycles per unit. This means that in the span of one unit along the x-axis, the cotangent function will complete \( \frac{1}{4} \) of its cycle.

Amplitude of Trigonometric Functions

In trigonometry, amplitude typically refers to the height of the peak (or depth of the trough) of the wave from its central axis. However, for cotangent and tangent functions, which do not have a maximum or minimum amplitude due to their infinitely increasing or decreasing nature, the concept of amplitude is slightly different.

Instead, we consider the 'amplitude' as the absolute value of the coefficient of the trigonometric function term. This influences the steepness of the wave but does not bound its height. For the function \( y = -2 \cot(\frac{\pi}{4}x) \), the amplitude is \( |-2| = 2 \), which means the graph's rate of change from the central axis will be steeper compared to the basic cotangent function. It's essential to note that while the cotangent function does not have a bounded amplitude like sine or cosine, this coefficient still affects how abruptly the function approaches its asymptotes.