Problem 20 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The period of \( y = \tan(x - \frac{\pi}{4}) \) is \( \pi \). Graph it by shifting \( y = \tan x \) right by \( \frac{\pi}{4} \).

Step 1: Identify the Standard Period of Tangent

The standard period of the tangent function, \(y = \tan(x)\), is \(\pi\). This means the function repeats every \(\pi\) units along the x-axis.

Step 2: Determine the Effects of Transformations

The given function is \(y = \tan(x - \frac{\pi}{4})\). The transformation \(x - \frac{\pi}{4}\) indicates a horizontal shift. The period is not affected by horizontal shifts, so it remains \(\pi\).

Step 3: Graph the Core Tangent Function

To graph \(y = \tan(x)\), identify key points: the vertical asymptotes occur at \(x = -\frac{\pi}{2} + n\pi\) where \(n\) is an integer, and the function passes through the origin (0,0).

Step 4: Apply the Horizontal Shift

Apply the transformation of shifting the graph \(\frac{\pi}{4}\) units to the right. This moves each point on the graph to a new position, including the midpoint and asymptotes.

Step 5: Mark Key Features and Draw the Full Period

With the shift, the vertical asymptotes are now at \(x = -\frac{\pi}{4} + n\pi\), and the midpoints where the function crosses the x-axis are shifted to start from \(\frac{\pi}{4}\) instead of 0. Draw one period of the function using these shifted points.

Key Concepts

Tangent Function

The tangent function, noted as \(y = \tan(x)\), is one of the basic trigonometric functions and is quite unique due to its repeated vertical asymptotes and rising nature. It originates from the ratio of sine and cosine functions: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This explains why the tangent function is undefined whenever \(\cos(x) = 0\).

This happens at \(x = \frac{\pi}{2} + n\pi\) where \(n\) is an integer, leading to vertical asymptotes. Furthermore, the tangent function has interesting properties:

• It is periodic and repeats its pattern every \(\pi\) units.
• The function starts at the origin (0,0) in its standard form and increases or decreases without bounds until reaching the next asymptote.
• For each period, it crosses the x-axis at a single midpoint, typically at integer multiples of \(\pi\).

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their basic shape, period, and key features like asymptotes and intercepts. For the tangent function:

• Begin by sketching the standard form \(y = \tan(x)\), marked with asymptotes at \(x = -\frac{\pi}{2} + n\pi\).
• The function crosses the x-axis every half period \(x = n\pi\), where it returns to traverse from minus to plus infinity past the next asymptote.

This visual representation allows us to identify transformations applied, such as shifts and stretches. By assessing these basics:

• Recognize how the graph is altered by transformations, such as phase shifts or amplitude changes.
• The importance of correctly positioning asymptotes cannot be overstated, as these are crucial for defining the sections where the function remains undefined.
• Ensure the curve smoothly passes through the midpoints for accuracy.

Function Transformation

Transformation involves altering a function to change its position or shape on the graph. Consider the transformation in \(y = \tan(x - \frac{\pi}{4})\):

• Horizontal Shift: The term \(-\frac{\pi}{4}\) indicates a rightward shift of \(\frac{\pi}{4}\).
• This transformation involves shifting every feature of the graph, including asymptotes and x-intercepts, exactly \(\frac{\pi}{4}\) units to the right.
• There are no vertical changes or stretches, so the look and vertical span of the tangent function stay the same.

Function transformations are essential for adjusting trigonometric graphs to model situations or fit data depending on the initial condition or given parameters.

The overall period or cycle length remains unaffected by horizontal shifts, maintaining its initial value.

Period of Functions

The period of a function is the distance on the x-axis over which the function's shape repeats. For the function \(y = \tan(x)\):

• Standard Period: This is \(\pi\), meaning the function repeats every \(\pi\) units.
• A period is crucial for pattern predictions which help us anticipate future graph behavior for ongoing cycles.

In transformations like \(y = \tan(x - \frac{\pi}{4})\), the period remains \(\pi\), despite the shift. Recognizing the consistent period helps in predicting interception points and asymptotes despite shifts:

• Clickback and forth by \(\pi\) to observe identical graph sections.
• Understand that while intercept and asymptote positions alter with shifts, repetition distance does not.

This consistent cyclical nature is key to understanding and managing repetitive trigonometric functions.