Problem 30 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(g(x)=4 \cot x\)

Step-by-Step Solution

Answer

The function \(g(x) = 4 \cot x\) is undefined for \(x = n \pi\), where n is an integer, and has a period of \(\pi\). It starts from positive infinity at \(x = 0\), decreases to 0 at \(x = \pi / 2\), decreases further to negative infinity at \(x = \pi\), and then jumps back to positive infinity at \(x = \pi + 0.01\). All its y-coordinates are multiplied by 4 as implied by the coefficient 4, meaning that its maximum and minimum are \(+\infty\) and \(-\infty\), respectively. The graph repeats its behavior every \(\pi\).

Step 1: Analyze the cotangent function

The cotangent function, written as \(\cot x\), is the reciprocal of the tangent function, i.e., \(\cot x = \frac{1}{\tan x}\). It is undefined for \(x = n \pi\), where n is an integer, and it has period \(\pi\). The function \(\cot x\) starts from positive infinity (\(+\infty\)) at \(x = 0\), decreases to 0 at \(x = \pi / 2\), then to negative infinity (\(-\infty\)) at \(x = \pi\), and then jumps back to positive infinity at \(x = \pi + 0.01\) and the cycle repeats every \(\pi\).

Step 2: Analyze the coefficient before the cotangent function

The coefficient before the cotangent function, in this case, is 4. Generally, a coefficient before a function can vertically stretch or compress the function, depending on whether the coefficient is more than 1 or less than 1. In this case, the coefficient is regular, meaning that the graph will be stretched vertically by a factor of 4. The vertical stretch doesn't affect the function's period, asymptotes, or its undefined points.

Step 3: Summarize the implied properties of \(g(x) = 4 \cot x\)

Taking into account the properties of the cotangent function and the vertical stretch implied by the coefficient, \(g(x) = 4 \cot x\) is undefined for \(x = n \pi\), where n is an integer. It has period of \(\pi\). The function starts from positive infinity at \(x = 0\), decreases to 0 at \(x = \pi / 2\), decreases further to negative infinity at \(x = \pi\), and then jumps back to positive infinity at \(x = \pi + 0.01\). All its y-coordinates are multiplied by 4 as implied by the coefficient 4, meaning that its maximum and minimum are \(+\infty\) and \(-\infty\), respectively. The graph repeats its behavior every \(\pi\).

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot x \), is a fundamental trigonometric function. It is specially known as the reciprocal of the tangent function. This means \( \cot x = \frac{1}{\tan x} \). It is a smooth curve which is defined everywhere except at multiples of \( \pi \), where the tangent function's value hits zero causing \( \cot x \) to become undefined. At these points, the cotangent function has vertical asymptotes, which means the function approaches infinity in both the positive and negative directions.

Important properties of the cotangent function include:

• Its graph always starts from positive infinity, decreases to zero, and continues to negative infinity until it resets at the next multiple of \( \pi \).
• The period is \( \pi \), meaning the function pattern repeats every \( \pi \) units along the x-axis.

This period of repetition and structure of asymptotic behavior gives the cotangent function its distinctive wave-like pattern in graphs.

Function Transformation

When we apply transformations to functions, it involves changing the function's graph in various ways, such as shifting, stretching, or reflecting. For example, in the case of \( g(x) = 4 \cot x \), we are particularly interested in the vertical stretching transformation.

A vertical transformation involves multiplying the function by a constant. This constant, known as a scaling factor, can stretch or compress the graph vertically. In \( g(x) = 4 \cot x \), the factor here is 4, meaning every y-coordinate of the cotangent function is multiplied by 4.

Key points about function transformation include:

• The scaling factor modifies the height of the peaks and troughs of the function without affecting its period or x-intercepts.
• The overall shape of the function stays intact; only the amplitude is modified.

This concept helps in understanding how combining transformations can affect the visual representation of basic functions in complex ways.

Periodicity

Periodicity is a vital characteristic of trigonometric functions. It refers to the repeating nature of these functions over regular intervals. For the cotangent function, the period is \( \pi \). This indicates that the function's behavior and graph repeat every \( \pi \) units on the x-axis.

Understanding periodicity is crucial because:

• It helps predict the function's values for any x by recognizing repeating patterns.
• It simplifies the analysis of the function's graph over broader domains without plotting every possible x-value.

In \( g(x) = 4 \cot x \), the periodicity remains unaffected by the vertical stretch. Thus, the pattern of ascending and descending from infinity to zero and back to infinity within the range of 0 to \( \pi \) is consistent throughout the graph.

Vertical Stretch

A vertical stretch occurs when we multiply a function by a factor greater than one. In \( g(x) = 4 \cot x \), the vertical stretch factor is 4. This stretching alters the magnitude of the function points, meaning that every point on \( \cot x \) is stretched vertically by a factor of 4.

The impact of a vertical stretch includes:

• The function's maximum and minimum values (conceptually) are magnified. However, in the case of \( \cot x \), these are always infinite.
• The function approaches its vertical asymptotes more steeply.

Despite these changes, the vertical stretch does not affect the function's inherent periodicity or x-axis interceptions. It purely modifies the steepness and height of the waves, making it a powerful tool in function transformation.