Problem 5 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of vertical translations to graph at least two cycles of the given functions. $$g(x)=\cot x+\frac{3}{2}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = \cot x + \frac{3}{2}\) looks like the graph of \(\cot x\), but shifted up by \(3/2\) units. Therefore, two cycles can be represented by the portions of the function between \(x = 0\) to \(x = 2\pi\), and \(x = -\pi\) to \(x = \pi\)

Step 1: Draw the Basic Shape of the Cotangent Function

The cotangent function, \(\cot x\), has a period of \(\pi\), with vertical asymptotes at \(x = n\pi\), where \(n\) is any integer. It starts from infinity at 0, decreases to 0 at \(\pi/2\), and then decreases to negative infinity at \(\pi\). This pattern repeats for every period. Therefore, as the first step, draw the basic form of \(\cot x\) for two periods, say, from \(x = 0\) to \(x = 2\pi\), and from \(x = -\pi\) to \(x = \pi\).

Step 2: Apply the Vertical Translation

The vertical translation applies uniformly at all points on the graph, shifting it up by \(3/2\) units. Therefore, go to every point on the graph drawn in step 1 and shift them up by \(3/2\) units. The asymptotes, being at infinity, remain unchanged.

Step 3: Finalize the Graph

After shifting every point from step 1 up by \(3/2\) units, we would get the graph of \(g(x) = \cot x + \frac{3}{2}\). Note that the general shape of \(\cot x\) is preserved, but it's now located \(3/2\) units higher. This graph represents two cycles of \(g(x)\) as requested.

Key Concepts

Understanding Vertical Translations

When graphing trigonometric functions, a vertical translation involves shifting the graph up or down on the coordinate plane. To visualize this, picture the original graph of a function as an elastic sheet. A vertical translation then corresponds to pulling this sheet up or pushing it down without altering its shape.

In the context of our exercise with the function \(g(x) = \cot x + \frac{3}{2}\), the \(\frac{3}{2}\) signifies a vertical translation. Each point on the original cotangent graph is lifted exactly \(\frac{3}{2}\) units upwards, in a parallel movement. The most important part here is to remember that vertical translations do not affect the period, amplitude or direction of the original function—only its position relative to the x-axis changes. Consequently, for students learning to graph such functions, it's critical to first understand the base function before applying any translations.

Cotangent Function Characteristics

The cotangent function, \(\cot x\), exhibits a set of distinct characteristics that set it apart from other trigonometric functions. Firstly, the cotangent function is the reciprocal of the tangent function, which means it can be defined as \(\cot x = 1/\tan x\). As a result, it has undefined values wherever the tangent function has zeros, leading to vertical asymptotes.

Key characteristics include:

• Period: The cotangent function repeats every \(\pi\) units, making \(\pi\) its period.
• Domain: All real numbers, except at multiples of \(\pi\), where it's undefined.
• Range: All real numbers can be cotangent values.
• Asymptotes: Present at \(x = n\pi\), where \(n\) is an integer, coinciding with zeros of the tangent function.
• Symmetry: Cotangent is an odd function, meaning \(\cot(-x) = -\cot x\).

Understanding these characteristics is essential for accurately plotting the function and applying transformations like vertical translations.

Asymptotes in Trigonometric Functions

Asymptotes represent the lines that a function's graph approaches but never touches or crosses. In trigonometric functions, such as the cotangent function, asymptotes occur due to the function growing infinitely large or decreasing infinitely small in value. These are known as vertical asymptotes and are integral to the shape of the graph.

For \(\cot x\), vertical asymptotes occur where the function is undefined - at every integer multiple of \(\pi\). As such, when graphing \(g(x) = \cot x + \frac{3}{2}\), these asymptotes remain unchanged despite the vertical shift because they represent points of infinite (or undefined) value—no finite translation can affect them.

It's important for students to carefully plot these asymptotes and recognize their unchanging nature during translations. They serve as a guide to drawing the rest of the function, ensuring that the curves of the function approach the asymptotes appropriately within each period.