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Chapter 1: Problem 3

$$ y=\frac{1}{2} \cot \pi x $$

Short Answer

Expert verified
The function \(y = \frac{1}{2} \cot \pi x\) represents a cotangent function that is vertically scaled by a factor of \(\frac{1}{2}\) and horizontally compressed by a factor of \(\pi\). It approaches infinity as \(x\) tends to 0 and is infinite at \(x = n\). At \(x = \frac{1}{2}\) and \(x = 1.5\), the function intersects the x-axis.

Step by step solution

01

Understanding the function

Firstly, the function \(y= \frac{1}{2} \cot \pi x\) needs to be understood. This function is a Cotangent function which is shifted and scaled due to the coefficients. The \(\frac{1}{2}\) is the vertical stretching factor and \(\pi\) is the horizontal compression factor.
02

Breaking down in simple cotangent function

The function can be transformed into simpler terms by putting \(\pi x = z\). This transforms the given function into \(y= \frac{1}{2} \cot z\).
03

Representing the function graphically

The cotangent function is periodic with a period of \(\pi\). It approaches infinity as z tends to 0 and is infinite at z = n\(\pi\). At z = \(\frac{\pi}{2}\) and \( \frac{3\pi}{2}\), the function intersects the x-axis.
04

Transforming back to the original function

Replacing \(z\) with \(\pi x\) transforms the function back to original form. It can be seen that the factor \(\pi\) causes a horizontal compression of the graph by a factor of \(\pi\) and the factor \(\frac{1}{2}\) causes a vertical scaling by factor \(\frac{1}{2}\). This means that all y-values are halved causing the graph to come closer to the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent Function
The cotangent function is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. Therefore, the cotangent of an angle in a right-angled triangle is the ratio of the adjacent side to the opposite side. In terms of the unit circle, it can be expressed as:\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]Unlike sine and cosine, cotangent is undefined when the sine of an angle is zero because division by zero is not possible. This characteristic results in vertical asymptotes in its graph, wherever the sine function equals zero.
  • Key Features: The cotangent function is decreasing in its principal interval of (0, \pi), crossing the x-axis at \( \frac{\pi}{2} \) and surrounding points.
  • Undefined Points: Due to being reciprocal of sine, it is undefined and has vertical asymptotes at integral multiples of 蟺.
Graph Transformation
In mathematics, graph transformations pertain to shifting, stretching, or compressing graphs of functions. The transformation of the cotangent function allows us to understand how parameters affect its graph.
  • Vertical Stretch/Compression: The coefficient \(\frac{1}{2}\) in \(y= \frac{1}{2} \cot(\pi x)\) scales the graph vertically by a factor of \(\frac{1}{2}\). This means all y-values are halved, bringing the peaks and troughs closer to the x-axis.
  • Horizontal Compression: The \(\pi\) in the argument compresses the graph horizontally. The entire function completes its cycle in a smaller range, effectively reducing the period.
These transformations adjust the graph's appearance significantly, changing both its amplitude and interval between zeros.
Periodicity
Periodicity is a fundamental property of trigonometric functions, which refers to the repetition of values at regular intervals. For the cotangent function, its natural period is \(\pi\), meaning it repeats its pattern every \(\pi\) units.
  • Naturally Periodic: The function \(\cot(z)\) returns to the starting point every \(\pi\). However, the presence of a factor like \(\pi x\) in the cotangent's argument compresses this period. Thus, the period of the function \(\frac{1}{2} \cot(\pi x)\) becomes \(\frac{1}{\pi} \cdot \pi = 1\).
  • Impact of Transformations: With transformations, while the periodic nature remains intrinsic, the length of each period might change. Horizontal stretching or compressing shifts the distances between repeating cycles.
Understanding periodicity assists in predicting function behavior over any continuous interval, a critical aspect for functions applied in oscillatory phenomena like waves or sound.

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Most popular questions from this chapter

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