Problem 39 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The period is \( \frac{\pi}{2} \), and the graph is shifted left by \( \frac{\pi}{2} \).

Step 1: Understanding the Function Form

The given function is in the form of a tangent function, which is typically represented as \( y = \tan(b(x - c)) \). For the given function \( y = \tan 2\left(x + \frac{\pi}{2}\right) \), we identify that \( b = 2 \) and \( c = -\frac{\pi}{2} \).

Step 2: Identifying the Period

The period of a basic tangent function \( y = \tan x \) is \( \pi \). For a function \( y = \tan(bx) \), the period is adjusted by dividing \( \pi \) by the absolute value of \( b \). Thus, the period of \( y = \tan 2\left(x+\frac{\pi}{2}\right) \) is \( \frac{\pi}{2} \), since \( b = 2 \).

Step 3: Identifying Shifts in the Graph

The function has a horizontal phase shift because of the \( x + \frac{\pi}{2} \) term. This represents a shift of \( -\frac{\pi}{2} \) units to the left. There's no vertical shift or change in amplitude for a tangent function.

Step 4: Graphing the Function

Start graphing by considering the standard tangent curve, which has vertical asymptotes where the function is undefined. For \( y = \tan 2x \), these occur at \( x = \frac{\pi}{4} + k\frac{\pi}{2} \) for integers \( k \). Apply the horizontal shift to each asymptote point, bringing them left by \( \frac{\pi}{2} \). This gives asymptotes at \( x = -\frac{\pi}{4} + k\frac{\pi}{2} \). Then, plot the tangent curve repeating every \( \frac{\pi}{2} \).

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions, often represented as \( y = \tan x \). Its graph is unique compared to sine and cosine functions, which are wave-like and continuous. Instead, the tangent function consists of repeated curves with vertical asymptotes, where the function is undefined. This characteristic creates a series of "S" shaped curves that repeat periodically across the graph.

The function value equals the sine of \( x \) divided by the cosine of \( x \). This can be crucial when analyzing its behavior, as the tangent is undefined wherever the cosine is zero. Therefore, at these points, the graph will have vertical asymptotes. Understanding the basic form of the tangent function helps in identifying how transformations affect its graph.

Period of a Function

The period of a function describes the interval over which it completes one full cycle before repeating. For the tangent function \( y = \tan x \), the basic period is \( \pi \), which is the distance between consecutive vertical asymptotes. This is different from sine and cosine, which have a period of \( 2\pi \).

When dealing with transformations, the formula to determine the new period of a tangent function \( y = \tan(bx) \) is \( \frac{\pi}{|b|} \). It means that changes in the variable \( b \) affect how quickly or slowly the function repeats. In the example \( y = \tan 2(x + \frac{\pi}{2}) \), \( b = 2 \), resulting in a period of \( \frac{\pi}{2} \). This indicates the function completes its repeating pattern more frequently compared to the standard tangent function.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric function's graph. In simpler terms, it tells us how much the function has moved left or right. For a general function like \( y = \tan(b(x - c)) \), the phase shift is calculated as \( c \).

Looking at \( y = \tan 2(x + \frac{\pi}{2}) \), the phase shift comes from the \( x + \frac{\pi}{2} \) term. Here, the graph shifts by \( -\frac{\pi}{2} \) units to the left, because having \( x + \frac{\pi}{2} \) is equivalent to \( x - (-\frac{\pi}{2}) \). Hence, every critical point, including the asymptotes, moves horizontally left by \( \frac{\pi}{2} \).

Graph Transformations

Graph transformations help us understand how changes in a function's equation affect its graph. For functions like the tangent, it involves shifts, stretches, and compressions.

With \( y = \tan 2(x + \frac{\pi}{2}) \), we observe two main transformations:

• **Horizontal Scaling**: The coefficient \( b = 2 \) compresses the graph horizontally, reducing the period to \( \frac{\pi}{2} \).
• **Horizontal Shift**: The term \( x + \frac{\pi}{2} \) creates a phase shift of \( -\frac{\pi}{2} \) to the left.

Together, these transformations influence how the graph of \( y = \tan x \) appears. Asymptotes change positions, and the pattern of the tangent curve repeats more frequently.