Problem 32 - Precalculus : Building Concepts and Connections

Question

Graph at least two cycles of the given functions. $$h(x)=2 \cos \left(2 x+\frac{\pi}{2}\right)-1$$

Step-by-Step Solution

Answer

The transformed cotangent function, \( h(x) = 2 \cot \left(x+\frac{\pi}{2}\right)-1 \), has a period of \( \pi \), vertical asymptotes at \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \) for one cycle, and key points shifted, stretched, and flipped accordingly. Sketch two or more cycles using this transformed period and key points.

Step 1: Identify the key points and periods of the cotangent function

The basic cotangent function \( \cot(x) \) has its key points at \( x = 0 \), \( x = \frac{\pi}{2} \), \( x = \pi \), and \( x = \frac{3\pi}{2} \). The function is undefined at \( x = 0 \) and \( x = \pi \), which are also the positions of the vertical asymptotes. With a period of \( \pi \), we graph two cycles from \( 0 \) to \( 2\pi \).

Step 2: Apply the horizontal shift

The fact that the function is \( h(x) = 2\cot\left(x+\frac{\pi}{2}\right)-1 \) means it's shifted \( \frac{\pi}{2} \) units to the left. Therefore, subtract \( \frac{\pi}{2} \) from the x-values of the key points and vertical asymptotes identified in step 1.

Step 3: Apply the vertical stretch and vertical shift

The coefficient of the function \( 2\cot\left(x+\frac{\pi}{2}\right) \) implies a vertical stretch by a factor of 2. So, multiply the y-values of the key points by 2. The constant term -1 means the function is shifted down by 1 unit. Thus, subtract 1 from the y-values.

Step 4: Plot the graph

Finally, plot the transformed key points and asymptotes. Extend the sketch to graph at least two cycles of the function going through the key points and separated by the asymptotes.

Key Concepts

Cotangent Function Characteristics

Understanding the cotangent function's characteristics is fundamental for graphing it accurately. The cotangent function, denoted as \( \cot(x) \), is the reciprocal of the tangent function and thus undefined where the tangent is zero - at \( x = 0 \) and \( x = \pi \). Its key characteristics include:

• Periodicity: The basic \( \cot(x) \) has a period of \( \pi \) - it repeats every \( \pi \) units along the x-axis.
• Asymptotes: Considering \( \cot(x) \) is undefined for certain values, it has vertical asymptotes at these points, which occur at every \( k\pi \) where \( k \) is an integer.
• Symmetry: The cotangent function is an odd function, meaning it is symmetric about the origin (\(180^\circ\) rotational symmetry).

These properties must be carefully incorporated when graphing the cotangent function, recognizing the periodic nature and the locations of vertical asymptotes is essential for accurate plots.

Vertical and Horizontal Shifts in Trigonometry

In trigonometry, both vertical and horizontal shifts play an integral role in transforming functions. A horizontal shift, sometimes called a phase shift, moves the graph left or right by a specific amount. This is represented by an addition or subtraction within the function's argument. For instance, \( h(x) = \cot(x + c) \) indicates a horizontal shift of \( c \) units to the left if \( c > 0 \) and to the right if \( c < 0 \).On the other hand, a vertical shift moves the graph up or down without altering its shape, described by adding or subtracting a constant outside the function. For the equation \( h(x) = \cot(x) + d \), the function moves \( d \) units upwards if \( d > 0 \) and downwards if \( d < 0 \) without changing the x-values. Mastering these shifts is crucial in graphing because it alters the position of key points and asymptotes, thereby changing the graph's overall appearance.

Transformation of Trigonometric Functions

Transformation of trigonometric functions is a blend of various modifications including stretching, shrinking, reflecting, and shifting, which can dramatically change the graph's outlook. When graphing transformed functions like \( h(x) = 2\cot(x + \frac{\pi}{2}) - 1 \), several steps are involved:

Horizontal and Vertical Shifts

Firstly, recognize the translation effect. In our exercise, \( x + \frac{\pi}{2} \) hints at a horizontal shift \( \frac{\pi}{2} \) units left, and the -1 indicates a vertical shift 1 unit down.

Stretching and Shrinking

Secondly, any coefficient in front of the \( \cot \) function, such as 2 in our example, signifies a vertical stretch, which multiplies the y-values of key points by that coefficient.Implementing these transformations step by step, as shown in the provided solution, allows for constructing an accurate graph. By stretching vertically, shifting both horizontally and vertically, the function's appearance and the placement of asymptotes and key points are affected and should be reflected on the newly graphed function.