Problem 23 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The period is \( \pi \), with a phase shift of \( -\frac{\pi}{4} \) to the left.

Step 1: Identify the Base Function

The given function is based on the cotangent function, which is \( y = \cot(x) \). The cotangent function has a period of \( \pi \).

Step 2: Determine the Period

The period of \( y = \cot(x) \) is \( \pi \). The function \( y = \cot \left( x + \frac{\pi}{4} \right) \) includes a horizontal shift but does not affect the period. Therefore, the period remains \( \pi \).

Step 3: Identify Transformation Effects

The argument \( x+\frac{\pi}{4} \) indicates a phase shift (horizontal shift) by \( -\frac{\pi}{4} \). This means the graph of the function is shifted \( \frac{\pi}{4} \) units to the left.

Step 4: Sketch the Graph

To graph \( y = \cot(x+\frac{\pi}{4}) \), start with the graph of \( y = \cot x \), which has vertical asymptotes at \( x = n\pi \), where \( n \) is an integer. Shift each point and asymptote to the left by \( \frac{\pi}{4} \) to account for the horizontal shift. The asymptotes will occur at \( x = n\pi - \frac{\pi}{4} \). The period remains \( \pi \).

Key Concepts

Cotangent Function

The cotangent function, denoted by \( \cot(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function, meaning \( \cot(x) = \frac{1}{\tan(x)} \). In terms of sine and cosine, it can also be expressed as:\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]This function is known for its characteristic wave form and periodic nature. Some key features of the cotangent function include:

• Vertical asymptotes at integer multiples of \( \pi \), i.e., at \( x = n\pi \), where \( n \) is an integer.
• It crosses the x-axis at points \( x = (2n+1)\frac{\pi}{2} \).
• It decreases moving from left to right.

Understanding the basic behavior of \( \cot(x) \) is crucial for graphing more complex variations and transformations of the function. This includes recognizing the locations of asymptotes and zeros, which are central in sketching its graph.

Graphing Transformations

Graphing transformations, like shifts and stretches, help us modify the basic trigonometric function to fit specific cases. For the given function \( y = \cot(x + \frac{\pi}{4}) \), we apply a transformation to the standard \( y = \cot(x) \) graph.A common transformation for trigonometric graphs is the horizontal shift, also known as the phase shift. In our function, the term \( x + \frac{\pi}{4} \) indicates a phase shift to the left by \( \frac{\pi}{4} \) units. This means every point on the graph moves left:

• The vertical asymptotes, originally at \( x = n\pi \), will now be located at \( x = n\pi - \frac{\pi}{4} \).
• The zeros of the function will also shift left by \( \frac{\pi}{4} \).

When transforming a graph, remember that these changes do not affect the shape or size of the graph, only its position. By mastering transformations, you can quickly sketch any trig function regardless of its added complexities.

Period of Functions

The period of a function is the distance over the x-axis it takes before the function begins to repeat. For trigonometric functions, this is a crucial attribute. The cotangent function, \( \cot(x) \), has a standard period of \( \pi \), meaning it repeats every \( \pi \) units.In the transformed function \( y = \cot(x + \frac{\pi}{4}) \), while there is a horizontal shift, it does not affect the function's period. The period remains \( \pi \). This means that every interval of \( \pi \) along the x-axis contains one complete wave of the cotangent function.Key points to remember about periods:

• Horizontal or phase shifts do not impact the period.
• The period of \( \cot(bx) \) is \( \frac{\pi}{|b|} \). In our scenario, \( b = 1 \), so the period is still \( \pi \).

Understanding period adjustments is important for predicting where the function will behave in its cyclic manner and is a foundational element when working with trigonometric transformations.