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Chapter 6: Problem 14

Graph each function over a one-period interval. $$y=-\cot \frac{1}{2} x$$

Short Answer

Expert verified
The one-period interval for \(y = -\cot \frac{1}{2} x\) is from \(\textbf{0 to 2蟺}\), with vertical asymptotes at \(\textbf{0 and 2蟺}\).

Step by step solution

01

Identify the basic cotangent function's properties

The cotangent function, \(y = \cot x\), has its period as \(\textbf{蟺}\) and undefined points at \(\textbf{0, 蟺, 2蟺, ...}\). Its graph decreases from positive infinity to negative infinity within one period.
02

Determine the transformation due to \(\frac{1}{2}x\)

The argument \(\frac{1}{2}x\) means that the function is horizontally stretched. Therefore, the period of \(y = \cot \frac{1}{2} x\) becomes \(\textbf{2蟺}\) (i.e., multiplied by a factor of 2).
03

Analyze the effect of the negative sign

The negative sign before the cotangent function indicates a reflection over the x-axis. Therefore, \(y = -\cot \frac{1}{2} x\) will reflect the cotangent function horizontally.
04

Identify critical points and asymptotes

Since the period is now \(\textbf{2蟺}\), the critical points will be at \(\textbf{0, 2蟺}\). The function has vertical asymptotes at \(\textbf{x = 0}\), beginning and end of the interval.
05

Draw the graph

Plot the graph within one period from x = 0 to x = 2蟺. Since it鈥檚 a reflected cotangent function, start from x = 0 and plot the curve downward approaching -鈭� to +鈭� asymptotically as it approaches x = 2蟺. Mark the interval and critical behavior on the graph.

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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, denoted as \(y = \cot x\), is a key trigonometric function that relates the angles of a right triangle to the ratios of two of its sides. It is defined as the reciprocal of the tangent function:
periodic functions
A periodic function repeats its values in regular intervals or periods. For trigonometric functions like the cotangent, this characteristic means they create wave-like patterns when graphed. The period of the basic cotangent function \(y = \cot x\) is \pi\. This interval is the length it takes for the function to start repeating itself.
function transformation
Transformations alter the appearance and properties of a function. Types of transformations include:
  • Horizontal Stretches and Compressions: Multiply the input variable \(x\) by a constant. For example, \cot \left( \frac{1}{2} x \right)\ stretches the graph horizontally, doubling the period.
  • Vertical Reflections: Multiply the entire function. For instance, \y = -\cot x\ reflects the cotangent graph over the x-axis.
asymptotes
Asymptotes are lines that the graph of a function approaches but never actually touches. For \(y = \cot x\), vertical asymptotes occur at points where the function is undefined, such as \x = 0, \pi, 2\pi,\ and so forth. This characteristic shapes the cotangent graph, guiding it to approach infinity near these lines.

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