Problem 23 - Algebra and Trigonometry

Question

In Exercises 17–24, graph two periods of the given cotangent function. $$ y=3 \cot \left(x+\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The graph of the given cotangent function has asymptotes at \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\). The cotangent graph begins at the first asymptote and goes through the x-axis halfway between the asymptotes. It then proceeds to the second asymptote. The graph is vertically stretched by a factor of 3. This process is repeated for the next period.

Step 1: Identify the Shift

The function has a horizontal shift to the left of \(\frac{\pi}{2}\). This shift is added directly to \(x\) in the function. The standard period for the cotangent function has asymptotes at \(x=0\) and \(x=\pi\). After the shift, the asymptotes will be at \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\).

Step 2: Identify the Stretch

The coefficient of the cotangent function is 3, which vertically stretches the graph. This means that when we draw the graph, we will multiply all \(y\)-values by 3.

Step 3: Sketch the Graph

Draw the x and y axes and identify the range. Cotangent function is undefined at multiples of \(\pi\) and ranges from \(-\infty\) to \(+\infty\). Draw vertical dashed lines (asymptotes) at \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\), One period of the cotangent graph falls between these asymptotes. To graph the function, draw it starting from the asymptote at \(x=-\frac{\pi}{2}\) going towards the second asymptote at \(x=\frac{\pi}{2}\) meeting the x-axis at the mid-point between the asymptotes. Stretch the graph by a factor of 3. Repeat the same process to graph the next period.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is a fundamental trigonometric function that relates to the tangent function. Specifically, it is the reciprocal of the tangent function, which means:

• \( \cot(x) = \frac{1}{\tan(x)} \)

This function is used to analyze periodic phenomena and is defined by its repeating pattern of peaks and valleys. The standard cotangent function has some unique characteristics:

• It is undefined at points where the sine function is zero, usually multiples of \( \pi \).
• The range is \((-\infty, \infty)\).
• One period of the cotangent function spans the interval \((0, \pi)\).

Cotangent graphs decline from left to right within each period, due to their nature as reciprocals to the tangent graph, which rises. Understanding its properties is crucial in trigonometry, especially when transforming its graph.

Vertical Stretch

A vertical stretch involves altering the amplitude of a function, which for trigonometric graphs means changing the steepness or height of the graph. When examining the cotangent function, a common transformation is multiplying the function by a constant factor, such as 3 in our exercise.The effect of a vertical stretch by a factor of 3 on the function \( y = \cot(x) \) can be described as:

• Every point on the graph moves three times further from the x-axis.
• The stretching does not alter the period or the locations of asymptotes but changes the y-values.

In practical terms, the graph becomes steeper. This means the function’s increase or decrease is more pronounced. This is relevant when considering the graph's appearance and behavior along different intervals.

Horizontal Shift

A horizontal shift occurs when the graph of a function is moved left or right along the x-axis. For the cotangent function in the exercise \( y=3 \cot \left(x+\frac{\pi}{2}\right) \), the shift is determined by the expression \( x+\frac{\pi}{2} \). In our case, a shift of \(-\frac{\pi}{2}\) means:

• The graph moves left by \( \frac{\pi}{2} \) units.
• The original asymptotes at \( x= 0 \) and \( x= \pi \) now become \( x= -\frac{\pi}{2} \) and \( x= \frac{\pi}{2} \).

This transformation does not affect the graph's shape or vertical characteristics but adjusts its position on the horizontal plane. Knowing how to apply horizontal shifts is essential in graph transformations, especially in understanding phase shifts in wave-like functions.

Trigonometric Asymptotes

Asymptotes are lines that a graph of a function approaches but never reaches. In trigonometry, the cotangent function, like its cousin the tangent function, has vertical asymptotes. These lines mark the values where the function is undefined. For a standard \( \cot(x) \) function:

• Asymptotes occur at multiples of \( \pi \), such as \( x=0 \), \( x= \pi \), \( x=2\pi \), etc.

In the exercise \( y=3 \cot \left(x+\frac{\pi}{2}\right) \), the horizontal shift has moved these asymptotes to new positions:

• Now they appear at \( x=-\frac{\pi}{2} \) and \( x=\frac{\pi}{2} \).

Asymptotes are critical because they guide the flow of the cotangent graph between periods. Recognizing where these asymptotes lie helps define the periodic structure and behavior of the function when graphing.