Problem 17 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=\tan \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The period is \(\pi\). Graph by shifting the basic tangent function left by \(\frac{\pi}{2}\).

Step 1: Understanding the Tangent Function

The basic function of the tangent is \(y = \tan(x)\), which has a period of \(\pi\). This means the function repeats every \(\pi\) units.

Step 2: Identify the Transformation

The given function is \(y = \tan\left(x + \frac{\pi}{2}\right)\). This function is a horizontal shift of the basic tangent function by \(-\frac{\pi}{2}\), which means it shifts the graph of the function to the left by \(\frac{\pi}{2}\). This shift does not change the period of the tangent function.

Step 3: Determine the Period

Since the horizontal shift does not affect the period and the period of the basic tangent function is \(\pi\), the period of \(y = \tan(x + \frac{\pi}{2})\) remains \(\pi\).

Step 4: Graph the Function

To graph \(y = \tan\left(x + \frac{\pi}{2}\right)\), start by plotting the basic tangent function \(y = \tan(x)\), which has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. Shift these asymptotes \(\frac{\pi}{2}\) to the left, resulting in new asymptotes at \(x = k\pi\). Draw the tangent curve between these vertical asymptotes, noting that the curve now crosses the x-axis at \(x = -\frac{\pi}{2} + k\pi\), with \(k\) as an integer.

Key Concepts

Tangent Function

The tangent function, represented as \( y = \tan(x) \), is one of the fundamental trigonometric functions. It is distinctively known for its periodic nature, with a period of \( \pi \), signifying that its pattern repeats every \( \pi \) units along the x-axis. Unlike sine and cosine functions, which are bounded between -1 and 1, the tangent function has no boundaries, and it extends from negative to positive infinity. This function exhibits vertical asymptotes at intervals where it is undefined, which occurs at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. These asymptotes are crucial features that create breaks in the graph. Connecting the points between these asymptotes creates the characteristic "wave-like" curve of the tangent function. Understanding the basic behavior of the tangent function lays a solid foundation for exploring trigonometric graph transformations.

Graphing Transformations

Graphing transformations involve modifying a function to shift, stretch, or compress its graph. When graphing the function \( y = \tan(x + \frac{\pi}{2}) \), we are dealing with a horizontal shift transformation. Specifically, adding \( \frac{\pi}{2} \) inside the function translates the graph to the left by \( \frac{\pi}{2} \) units. This transformation is essentially a shift in the x-values, affecting where the wave-like sections and asymptotes of the function appear on the graph.

To apply a horizontal shift to \( y = \tan(x) \):

• The asymptotes, initially at \( x = \frac{\pi}{2} + k\pi \), shift to new positions at \( x = k\pi \).
• The graph crossings, previously at \( x = 0 + k\pi \), move to \( x = -\frac{\pi}{2} + k\pi \).
• The shape and orientation of the tangent curve remain unchanged, focusing solely on horizontal placement.

This method of transformation maintains the periodicity and shape of the original graph, allowing for an intuitive understanding of how functions in trigonometry can alter spatially.

Periodicity in Trigonometry

Periodicity is a crucial concept in trigonometry, referring to the repeating nature of trigonometric functions over a specific interval called the period. For the tangent function \( y = \tan(x) \), this period is \( \pi \), meaning the function repeats its wave pattern every \( \pi \) units. This property results in a predictable graph structure, crucial for calculating angular measures and cyclic phenomena in various fields like physics and engineering.

Understanding the periodicity of trigonometric functions such as tangent helps us predict their behavior:

• Each cycle of the tangent function's period includes a single rise and fall, marked with one vertical asymptote separating two sections.
• Periodicity allows the graph to be reconstructed from any initial point, just by knowing the period and pattern.
• Despite horizontal transformations, the period remains the same, reaffirming the inherent periodic behavior.

Mastery of periodicity simplifies the analysis of trigonometric functions, enhancing problem-solving capabilities in scenarios involving cyclical trends and patterns.