Problem 67 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi .\) Describe any phase shift and vertical shift in the graph. $$ y=\cot 2(x+\pi)+3 $$

Step-by-Step Solution

Answer

The function \(y=\cot 2(x+\pi)+3\) shifted left by \(\pi\) units (phase shift) and upwards by 3 units (vertical shift).

Step 1: Identify the phase shift and vertical shift

In the given cotangent function \(y=\cot 2(x+\pi)+3\), we can see that the phase shift value is represented by the quantity inside the parenthesis with the variable \(x\). In this case, \(x\) is shifted by \(-\pi\), which means that our function has a phase shift of \(-\pi\). The vertical shift is represented by the constant added at the end of the function, which in our case is 3, indicating a vertical shift upwards by 3 units.

Step 2: Determine the critical points

Before graphing the function, it is important to determine the critical points (undefined values and turning points) of the cotangent function. The cotangent function is undefined at \(x=n\pi\) for \(n\) is an integer due to the fact that the cotangent is defined as the reciprocal of the tangent which is zero at these points. Turning points occur at \(x=n\pi + \pi/2\). However, for our function these critical points will be shifted due to the phase shift of \(-\pi\). Therefore, undefined values and turning points of our function occur at \(2x=-n\pi-\pi\) and \(2x=-n\pi-\pi+\pi/2\), respectively.

Step 3: Graph the function

Now that we have evaluated the critical points for our function, the graph can be plotted over the interval [0, 2\(\pi\)]. The function will have vertical asymptotes (undefined points) at even multiples of \(\pi/2\) and will cross the y-axis (turning points) at odd multiples of \(\pi/2\). Since our function is shifted upwards by 3 units, these points and asymptotes will be drawn 3 units higher.

Key Concepts

Graphing Functions

Graphing trigonometric functions, like the cotangent function, involves understanding their general shape and behavior. Functions can be thought of as mathematical machines that take an input (x-value) and produce an output (y-value). The graph is essentially a visual representation of all the relationships between these inputs and outputs.
For cotangent functions, specifically, this involves recognizing features such as periodicity, asymptotes, and turning points. When graphing any function, these features must be accurately plotted based on their derived values.
To graph the function \( y = \cot 2(x + \pi) + 3 \), begin by identifying critical points and asymptotes, then adjust for phase and vertical shifts. This step-by-step approach ensures that you properly display all aspects of the function involved.

Phase Shift

Phase shift is an important concept that involves shifting the entire graph of a function horizontally. For the function \( y = \cot 2(x + \pi) + 3 \), the phase shift is found by examining the expression inside the parentheses. It shows how much the graph is shifted left or right along the x-axis.
In this problem, we have \( x + \pi \), which means there is a phase shift of \(-\pi\) units to the right. This shift changes where the periodic pattern of the cotangent function begins. Understanding phase shift is crucial for accurately plotting the starting points of functions and their movement on the graph.
Remember, a negative phase shift moves the graph to the left, while a positive shift moves it to the right. This affects the positions of all critical points and asymptotes of the function.

Vertical Shift

A vertical shift moves the entire graph up or down along the y-axis. This shift affects how the function appears in relation to the x-axis, without altering its shape or periodicity.
For the function \( y = \cot 2(x + \pi) + 3 \), the \(+3\) indicates a vertical shift upwards by 3 units. This means each point on the original cotangent graph moves up by 3 units. It directly impacts the y-values by adding 3 to every point.
Vertical shifts are visually easier to grasp than phase shifts but are equally important for ensuring the graph's accuracy. When graphing, first account for the vertical shift before plotting turning points and asymptotes. This ensures that every aspect of the function properly reflects the intended modifications.

Cotangent Function

The cotangent function \( y = \cot(x) \) is a trigonometric function that is the reciprocal of the tangent function, defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). This leads to certain unique characteristics in its graph.
The function is undefined where the sine of x is zero, which occurs at integer multiples of \( \pi \). These are where the vertical asymptotes are located. The function itself is periodic, repeating its behavior every \( \pi \) units.
Understanding the basic nature of the cotangent function, such as where it is undefined and its repeating pattern, is essential for correctly graphing any transformed version of it, such as \( y = \cot 2(x + \pi) + 3 \). When approaching cotangent functions, always consider adjustments like stretches, shifts, and reflections, which define the view and key characteristics of its graph.