Question

\(g(x)=3 \cot \frac{1}{4} x\)

Short answer

The function \(g(x)=3 \cot \frac{1}{4} x\) has an amplitude of 3 and a period of \(4\pi\). The function repeats every \(4\pi\), with vertical asymptotes at \(4\pi k\) for all integers \(k\), and has maximum values of \(3\), and minimum values of \(-3\).

Step-by-step solution

Identify the Period

For any trigonometric function, the period is defined as the distance over which the function repeats. For the cotangent function, the period is \(\pi\), but in this case, the function is stretched by a factor of \(\frac{1}{4}\), so its period is \(4\pi\).

Identify the Amplitude and Shift

The amplitude (maximum absolute value) is defined as the constant multiplying the cotangent function, which in this case is 3. There is no shift in this function because there is no constant being added or subtracted to the \(\frac{1}{4}x\) inside the cotangent.

Plot Key Points

Usually, the cotangent function takes on values from negative infinity to positive infinity, which determine the vertical asymptotes. In this case, due to the stretch, asymptotes occur every \(4\pi\), so plot points at \(2\pi,4\pi,6\pi\) and so on. Between these points, sketch in the shape of a cotangent function, with the function approaching positive or negative infinity as it nears an asymptote.

Factor in the Amplitude

The amplitude of the graph is 3, so the range of the graph will be \(-3\) to \(3\) between successive asymptotes. Graph the maximum points of the function at these heights.

Key concepts

Cotangent Function

The cotangent function, often written as \( \cot(x) \), is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function. Mathematically, it is expressed as \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \). This function is unique because it has both positive and negative infinity as its limits, creating vertical asymptotes.

• The cotangent function has no maximum or minimum points since it can extend infinitely in both directions.
• It is undefined where the sine function equals zero, creating vertical asymptotes.
• These vertical asymptotes occur at integer multiples of \( \pi \).

When graphing \( g(x) = 3 \cot \frac{1}{4} x \), these characteristics are influenced by the transformations applied. This function includes a stretch of \( \frac{1}{4} \), affecting its period and altering the spacing of vertical asymptotes.

Amplitude and Period

The amplitude and period are two critical characteristics when analyzing and graphing trigonometric functions. Although amplitude is often associated mainly with sine and cosine functions, here it defines the stretch factor in our transformed cotangent function.

• Amplitude: This refers to the maximum value a function attains. For the function \( g(x) = 3 \cot \frac{1}{4} x \), the amplitude is 3, which stretches the vertical distance the function covers.
• Period: The period of the cotangent function is the interval in which the function completes one full cycle. Normally, \( \cot(x) \) has a period of \( \pi \). However, due to the transformation by \( \frac{1}{4} \), the period extends to \( 4\pi \). Thus, the battles between cycles expand, influencing the separation between vertical asymptotes.

No horizontal shift occurs here, but understanding these transformations helps in predicting and plotting the function accurately.

Graphing Trigonometric Functions

Graphing a trigonometric function involves understanding its transformations and key points like asymptotes. For the function \( g(x) = 3 \cot \frac{1}{4} x \), each aspect of transformation affects its graphical representation:

• The vertical asymptotes: For \( g(x) \), they are repositioned based on its period \( 4\pi \). Asymptotes appear at \( 2\pi, 4\pi, 6\pi, \ldots \).
• The shape between asymptotes: The cotangent's "S" curve is transformational, extending its steepness due to the amplitude of 3.
• The vertical stretch: With an amplitude of 3, each curve stretches three units high and low, influencing how steep the curve appears within the period \( 4\pi \).

Plotting critical points helps sketch the function. As the function approaches each asymptote, it dramatically rises or falls, evidencing the distinctive nature of the cotangent curve. Practicing these plotting steps makes understanding trigonometric graphs easier.