Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed. $$ y=2+\frac{1}{6} \cot 2 x $$

Step-by-Step Solution

Answer

Period: \(\frac{\pi}{2}\), no amplitude, vertical shift up 2 units.

Step 1: Identify the Standard Form

The function is given as \[ y = 2 + \frac{1}{6} \cot 2x \] The standard form for a cotangent function is: \[ y = A + B \cot(Cx - D) \] where \(A\) is the vertical shift, \(B\) modifies the amplitude/vertical stretch, \(C\) affects the period, and \(D\) gives the phase shift.

Step 2: Determine the Amplitude and Vertical Stretch

For cotangent functions, there is no traditional amplitude as for sine or cosine. Instead, \(B = \frac{1}{6}\) indicates the vertical stretch. The graph is compressed vertically by a factor of \(\frac{1}{6}\).

Step 3: Calculate the Period

The period of the cotangent function is given by \[ P = \frac{\pi}{|C|} \] Here, \(C = 2\), so \[ P = \frac{\pi}{2} \] This indicates that one full cycle of the function occurs every \(\frac{\pi}{2}\) units on the x-axis.

Step 4: Determine the Phase Shift

The function does not include a \(D\) value (inside the cotangent). As such, there is no horizontal phase shift in this function.

Step 5: Determine the Vertical Shift

The vertical shift \(A = 2\). This shifts the graph upwards by 2 units.

Step 6: Summary of Transformations

The period is \(\frac{\pi}{2}\), there is a vertical stretch factor of \(\frac{1}{6}\), a vertical shift of 2 units upwards, and no horizontal shift.

Step 7: Sketching the Graph

To sketch the graph from \(-5 \leq x \leq 5\): 1. Mark the vertical shift by moving the centerline to \(y = 2\).2. Note that the period is \(\frac{\pi}{2}\). Use multiples of \(\frac{\pi}{2}\) to determine where each cycle starts and ends.3. The vertical stretch modifies the steepness of the graph. Normally, \(\cot x\) has a period of \(\pi\) with undefined points at \(n\pi, n \in \mathbb{Z}\). However, with the transformations, these points now occur at \(\frac{n\pi}{2}\).4. Plot several periods within the interval, marking points where the cotangent is undefined and intercepts.

Key Concepts

Cotangent Function

The cotangent function, often abbreviated as "cot", is one of the basic trigonometric functions. It is defined as the reciprocal of the tangent function:

• defined as \\( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \).

This function has a unique shape among trigonometric functions. Cotangent will have vertical asymptotes where the sine function is zero, leading to undefined values. These typically occur at integer multiples of \(\pi\), i.e., \(x = n\pi\), where n is an integer.
Since we're working with a transformed cotangent function in the exercise, recognize that transformations can include shifts, stretches, or compressions, affecting the graph's appearance significantly.

Graph Transformations

Graph transformations change the basic shape and position of a function's graph. For the function \(y = 2 + \frac{1}{6} \cot 2x\), several transformations are applied:

• Vertical Shift: \(+2\) moves the entire graph up by two units.
• Vertical Stretch: The factor \(\frac{1}{6}\) compresses the graph vertically, making it less steep.
• No Phase Shift: Since there is no 'D' term inside the cotangent, the graph does not shift horizontally.

These transformations help customize the graph, offering various visual representations. Such transformations are essential in adjusting functions to model real-world data accurately.

Periodicity

Periodicity refers to the way a function repeats its values at regular intervals. For trigonometric functions like cotangent, this periodic behavior is central.

• A standard \(\cot x\) function has a period of \(\pi\). This means it repeats every \(\pi\) units.

For our transformed function \(y = 2 + \frac{1}{6} \cot 2x\), the period changes due to the multiplier 2 in front of x. Calculate it by using the formula: \[P = \frac{\pi}{|C|}\] Where \(C = 2\). This results in:\[P = \frac{\pi}{2}\] The function now repeats every \(\frac{\pi}{2}\) units along the x-axis, leading to more frequent cycles compared to a standard cotangent graph.

Amplitude and Vertical Stretch

Unlike sine and cosine functions, cotangent does not have a traditional amplitude since it can approach infinity and doesn't oscillate between fixed maximum and minimum values.

• Instead, we consider vertical stretch or compression, characterized by "B" in the function's general form \(y = A + B \cot(Cx - D)\).

In the function \(y = 2 + \frac{1}{6} \cot 2x\), \(B = \frac{1}{6}\) means each value of the standard cotangent function is scaled down, making it less steep vertically.
This is crucial when drawing the function as it dictates how stretched or compressed the graph appears, especially between points of definition and vertical asymptotes.