Problem 18 - Algebra and Trigonometry

Question

7–52 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 \( \pi \); it is a phase shift right by \( \frac{\pi}{4} \).

Step 1: Identify the Function

The function given is a transformation of the basic tangent function, \( y = \tan x \), with a horizontal shift of \( \frac{\pi}{4} \) units to the right: \( y = \tan \left(x - \frac{\pi}{4}\right) \).

Step 2: Determine the Period of the Function

The period of the basic tangent function, \( y = \tan x \), is \( \pi \). Horizontal shifts do not affect the period of the function, so the period of \( y=\tan \left(x-\frac{\pi}{4}\right) \) remains \( \pi \).

Step 3: Identify the Horizontal Shift

The term \( x - \frac{\pi}{4} \) indicates a horizontal shift of \( \frac{\pi}{4} \) units to the right. This means all vertical asymptotes, intercepts, and points of \( y = \tan x \) are shifted accordingly.

Step 4: Create a Table of Key Values

Select points around one period (\( -\frac{\pi}{2} + \frac{\pi}{4} \) to \( \frac{\pi}{2} + \frac{\pi}{4} \)):- At \( x = \frac{\pi}{4} \), \( y = 0 \).- At \( x = 0 \), \( y = -1 \).- At \( x = \frac{\pi}{2} \), \( y = \infty \) (vertical asymptote).- At \( x = -\frac{\pi}{4} \), \( y = 1 \).- At \( x = -\frac{\pi}{2} \), \( y = -\infty \) (vertical asymptote).

Step 5: Graph the Function

Plot the key points identified in Step 4 and draw the curve of \( y = \tan \left(x - \frac{\pi}{4}\right) \).Make sure to include vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = -\frac{\pi}{2} \), and note the intercept at the point where the function crosses the x-axis.

Key Concepts

Trigonometric Functions

When exploring trigonometric functions, the tangent function, denoted as \( y = \tan x \), is fundamental. It relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. Unlike sine and cosine functions, which produce wave-like curves, tangent's graph exhibits periodic vertical asymptotes and has a distinct 'zigzag' pattern.

• The tangent function is periodic and not bounded, meaning it continues indefinitely in both directions.
• Its graph has vertical asymptotes at certain intervals, where the function approaches infinity.
• One functional unit of the tangent graph corresponds to an arc of \( \pi \) due to these asymptotes, repeated along the x-axis.

Understanding transformations of \( \tan x \), such as shifts, is crucial in analyzing more complex functions like \( y = \tan \left(x - \frac{\pi}{4}\right) \). Recognizing how these shifts impact the graph helps better visualize and interpret the function.

Period of a Function

The period of a function is the interval over which it completes one full cycle before repeating itself. For the tangent function \( y = \tan x \), this period is \( \pi \).

• The function repeats its unique pattern every \( \pi \) units on the x-axis.
• Horizontal shifts, such as \( x - \frac{\pi}{4} \), do not change the period.
• Vertical shifts, amplitude adjustments, and reflections can alter other characteristics of a function but not the period of the tangent function.

Knowing the period allows us to predict the behavior of a function across the x-axis, both within and outside the basic interval. For the function \( y = \tan \left(x - \frac{\pi}{4}\right) \), the periodicity \( \pi \) helps identify consistent patterns in its graph, crucial for accurate plotting and analysis.

Graphing Trigonometric Functions

Graphing trigonometric functions, like the tangent function, involves understanding key features such as periods, asymptotes, and shifts. Start by marking points where the function has asymptotes, approximating infinity. For \( y = \tan \left(x - \frac{\pi}{4}\right) \):

• The vertical asymptotes, shifted by \( \frac{\pi}{4} \), occur at \( x = \frac{\pi}{2} \) and \( x = -\frac{\pi}{2} \).
• The graph intercepts at \( y = 0 \), first occurring at \( x = \frac{\pi}{4} \), shifted right.
• Other significant points, such as \( x = 0 \) where \( y = -1 \), aid in sketching the curve.
• Vertical asymptotes help highlight one period's boundaries, while intercepts guide the curve path between these asymptotes.

Drawing these features ensures clarity in representing the function's periodic nature and any transformations. Once the essential points and asymptotes are plotted, connect them smoothly to reflect the tangent's characteristic pattern. Practicing these steps for varied functions enhances graph comprehension and accuracy.