Problem 73 - Algebra 2

Question

a. Graph \(y=\cot x, y=\cot 2 x, y=\cot (-2 x),\) and \(y=\cot \frac{1}{2} x\) on the same axes. b. Make a Conjecture Describe how the graph of \(y=\cot b x\) changes as the value of \(b\) changes.

Step-by-Step Solution

Answer

The graph of \(y=\cot bx\) changes based on the value of \(b\). For \( |b| > 1\), the graph compresses horizontally; for \(0 < |b| < 1\), it stretches horizontally. When \(b\) is negative, the graph reflects over the y-axis.

Step 1: Graph the cotangent functions

Start with graphing the functions \(y=\cot x\), \(y=\cot 2x\), \(y=\cot (-2x)\), and \(y=\cot (0.5x)\). The cotangent function has a period of \(\pi\) and all cotangent functions move between positive and negative infinity across their range. Note the points where the function is undefined (multiples of \(\pi\)) and observe how the function changes across each period. The cotangent function starts from positive infinity at \(0\) and moves to negative infinity at \(\pi\).

Step 2: Observe the changes for different \(b\) values

Notice that for \(y=\cot 2x\), the frequency of the waves doubles. It's a transformation that squishes the cotangent function horizontally by a factor of \(0.5\). For \(y=\cot -2x\), the cotangent function is mirrored vertically and horizontally, due the negative sign. For \(y=\cot 0.5x\), the function is wider, i.e., it stretches the function horizontally by a factor of \(2\). This is because the \(0.5\) in the argument of the cotangent function performs a horizontal stretch by a factor of \(1/0.5 = 2\).

Step 3: Conjecture on the transformation with variable \(b\)

Based on the graph transformations observed, one could make a conjecture about how modifying \(b\) in the formula \(y=\cot bx\) changes the graph. If \( |b| > 1\), the graph compresses horizontally by a factor of \(1/|b|\) (the frequency of the wave increases). If \( 0 < |b| < 1\), the graph stretches horizontally by a factor of \(1/|b|\) (the frequency of the wave decreases). If \(b < 0\), the graph reflects along the y-axis.

Key Concepts

Graph Transformations

When dealing with cotangent functions, graph transformations are key in understanding how the graph's shape and orientation adjust. Transformations can occur horizontally or vertically, depending on the modifications applied to the function. For instance, imagine adjusting the base graph of the cotangent function, which naturally begins at positive infinity and descends to negative infinity over its period. Here's how transformations change this pattern:

• Horizontal compression/stretching: This is affected by changing the value within the argument of the cotangent, such as in \( \cot 2x \) versus \( \cot \frac{1}{2}x \). A higher multiplier indicates more cycles within the same interval.
• Vertical reflection: Occurs when the function's argument has a negative multiplier, flipping the graph around the y-axis. An example is \( \cot(-2x) \), which also implies horizontal mirroring.

Graph transformations form an essential part of understanding how trigonometric functions, like the cotangent, behave under different circumstances.

Trigonometric Functions

Trigonometrics, such as the cotangent function, are part of the foundational tools in math used to describe angles and model periodic phenomena. Specifically, the cotangent is the reciprocal of the tangent function and is defined as: \[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \].
This function can span its range between positive and negative infinity while maintaining periodic zeros at intervals where sine equals zero (multiples of \( \pi\)).
The standard period of the cotangent function is \( \pi\), repeating its cycle as \(-\infty\rightarrow\infty\).

• Properties: As an odd function, it is symmetric regarding the origin, meaning \( \cot(-x) = -\cot(x) \).
• Asymptotes: Cotangent has vertical asymptotes wherever it is undefined, visually appearing every \(\pi\) units, such as at \(x = 0, \pi, 2\pi\), etc.

Understanding these characteristics enables us to predict how cotangent functions will look or act when graphically represented.

Frequency and Period Adjustments

One of the core changes in trigonometric functions relates to their frequency and period, particularly when adjustments are made to the argument of the function. For the cotangent function with equation \( y = \cot(bx) \), each adjustment in \( b \) produces significant visual changes in the graph:

• Frequency: Determines how often the cycle repeats within a given interval. By increasing \( |b| > 1\), we intensify the frequency, compressing the graph horizontally, resulting in more oscillations within a set range.
• Period: The distance between the repeating cycles. This is influenced inversely by the constant \( b \), as the period of \( y = \cot(bx) \) becomes \( \frac{\pi}{|b|} \). Thus, as \( |b| \) increases, the period diminishes by the factor of the reciprocal of \( |b| \).
• Reflection influence: When \( b \) is negative, the graph not only reflects about the y-axis but also compresses or stretches, altering both period and orientation.

Adjusting these two parameters allows detailed control over the function's graphical representation, showing various mathematical behaviors.