Problem 21 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=\cot \left(x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The period of the function is \( \pi \), and the graph is a left-shifted cotangent by \( \frac{\pi}{4} \).

Step 1: Understand the Function

The function given is the cotangent function with a phase shift. The function is written as \( y = \cot(x + \frac{\pi}{4}) \). This is a transformation of the basic cotangent function \( y = \cot x \).

Step 2: Determine the Basic Period of Cotangent

The basic period of the cotangent function \( y = \cot x \) is \( \pi \). This is because the cotangent function repeats every \( \pi \) units.

Step 3: Identify the Phase Shift

The function \( y = \cot(x + \frac{\pi}{4}) \) includes a phase shift. The phase shift is determined by the term \( +\frac{\pi}{4} \). This means the entire graph of \( y = \cot x \) is shifted to the left by \( \frac{\pi}{4} \).

Step 4: Calculate the Period of the Function

Since the cotangent function has no changes in the coefficient of \( x \), the period remains \( \pi \). The phase shift does not affect the period.

Step 5: Graph the Function

To graph \( y = \cot(x + \frac{\pi}{4}) \), take the basic cotangent function and shift every point to the left by \( \frac{\pi}{4} \). The main features of the graph occur at points shifted left by \( \frac{\pi}{4} \), such as the vertical asymptotes and midpoint crossings.

Key Concepts

Period of Trigonometric Functions

The period of a trigonometric function is the length of the smallest interval over which the function repeats itself. For basic trigonometric functions like sine, cosine, and tangent, understanding the period helps in predicting the behavior of their graphs.

For the cotangent function, the defined period is different from sine and cosine. While sine and cosine have a period of \(2\pi\), the cotangent function, \(y = \cot x\), repeats every \(\pi\) units. This means one complete cycle of the cotangent function occurs over a distance of \(\pi\) on the x-axis.

• The period is determined by the horizontal stretch or compression. However, in our exercise \(y = \cot(x + \frac{\pi}{4})\), there are no such transformations affecting the coefficient of \(x\).
• The period remains \(\pi\), unchanged by any phase shift.

Knowing the original period helps you determine where vertical asymptotes and zeros will occur on the graph.

Phase Shift

A phase shift refers to the horizontal movement of the graph of a trigonometric function. This shift is caused by adding or subtracting a constant to the variable inside the function.

In the equation \(y = \cot(x + \frac{\pi}{4})\), the term \(+\frac{\pi}{4}\) inside the cotangent function causes the entire graph to shift to the left by \(\frac{\pi}{4}\) units. This shift affects the positioning of key features:

• Vertical asymptotes, which are the lines where the function is undefined, move to the left.
• The zeros or x-intercepts are also shifted left by \(\frac{\pi}{4}\) units.

The phase shift, while affecting the position of these features, does not alter the function's period. Recognizing a phase shift is crucial for accurately locating and plotting the main traits in your graph.

Cotangent Function

The cotangent function, often written as \(y = \cot x\), is the reciprocal of the tangent function. Unlike sine and cosine which start their cycle at zero, cotangent begins with a characteristic undefined value at the origin. This creates distinct vertical asymptotes throughout its graph.

Some key features of the cotangent function include:

• Vertical Asymptotes occur at every integer multiple of its period, \(\pi\), since cotangent is undefined when the tangent is zero.
• The intervals between these asymptotes are where cotangent changes its values smoothly crossing zero at each midpoint.

The variation in the graph of \(y = \cot x\) is essential to understanding the horizontal shifts we apply. In our exercise, the graph experiences a phase shift due to the \(+\frac{\pi}{4}\) added inside the function.

This allows us to predict and plot the new positions of both the vertical asymptotes and the zeros for the transformed function \(y = \cot(x + \frac{\pi}{4})\). Understanding these characteristics of the cotangent function helps in visualizing its horizontal and vertical transformations accurately.