Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) period, (b) phase shift (if any), and (c) range of each function. $$y=-3 \tan \left[\frac{1}{2}\left(x+\frac{\pi}{4}\right)\right]$$

Step-by-Step Solution

Answer

Period: \(2\pi\); Phase Shift: \(-\frac{\pi}{4}\) left; Range: \((-\infty, \infty)\).

Step 1: Determine the Period of the Tangent Function

The general formula for the period of a tangent function, which is given by \( y = a \tan(bx + c) \), is \( \frac{\pi}{|b|} \). Here, \( b = \frac{1}{2} \). So, the period is \[ \frac{\pi}{\frac{1}{2}} = 2\pi. \]

Step 2: Find the Phase Shift

The phase shift of a tangent function is given by \( \frac{-c}{b} \), where the function is \( y = a\tan(bx + c) \). In this case, \( c = \frac{1}{2} \cdot \frac{\pi}{4} = \frac{\pi}{8} \). Therefore, the phase shift is \( \frac{-\frac{\pi}{8}}{\frac{1}{2}} = -\frac{\pi}{4} \). This indicates a shift to the left by \( \frac{\pi}{4} \).

Step 3: Determine the Range of the Tangent Function

The range of the tangent function is always all real numbers, \( (-\infty, \infty) \), regardless of the transformations made. This is due to the vertical asymptotes and its periodic nature.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine and cosine of an angle: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This definition implies that the tangent is undefined whenever the cosine is zero, leading to vertical asymptotes at those points. Unlike sine and cosine, the tangent function does not oscillate between fixed values but passes through all real numbers in a periodic manner.

The graph of the tangent function features repeating cycles, characterized by vertical asymptotes and "S" shaped curves. Each cycle is symmetrical around its vertical asymptotes and zero points. This behavior is due to the undefined nature of the function at certain points created by the division of zero, inherent in the tangent's definition.

• Equation: \( y = a \tan(bx + c) \)
• Key characteristics: asymptotes, periodicity

By understanding the basic graph and properties, you can then apply transformations like scaling, shifting, and reflecting to model various real-world phenomena.

Period of a Function

In the case of trigonometric functions like tangent, the period refers to the length of one complete cycle of the function. For the standard tangent function \( y = \tan(x) \), the period is \( \pi \). This means that the function pattern repeats every \( \pi \) units.

However, when modifying the function as \( y = a \tan(bx + c) \), the period changes inversely to the coefficient \( b \). The new period is calculated using \( \frac{\pi}{|b|} \). In the given exercise, \( b = \frac{1}{2} \), so the period becomes \( \frac{\pi}{\frac{1}{2}} = 2\pi \).

• The smaller \( |b| \), the longer the period.
• The larger \( |b| \), the shorter the period.

Understanding the concept of period is crucial for predicting the behavior of the function over different intervals.

Phase Shift

Phase shift in trigonometric functions, particularly for tangent, describes the horizontal movement of the graph along the x-axis. For the function \( y = a \tan(bx + c) \), the phase shift is determined by \( \frac{-c}{b} \).

Using the given equation, we find \( c = \frac{\pi}{8} \) and \( b = \frac{1}{2} \), so the phase shift is \( \frac{-\frac{\pi}{8}}{\frac{1}{2}} = -\frac{\pi}{4} \). This implies the whole function shifts \( \frac{\pi}{4} \) units to the left.

• A positive phase shift moves the graph to the left.
• A negative phase shift moves the graph to the right.

The phase shift can significantly alter the starting point of the periodic cycle, impacting where the function begins to rise or fall.

Range of a Function

The range of a function refers to the set of possible output values. For the tangent function, the range is unique because it spans all real numbers, \( (-\infty, \infty) \).

This is different from sine and cosine functions, which are limited to outputs between -1 and 1 due to their oscillatory nature. The unbounded range of the tangent results from its inability to have a maximum or minimum value. As the input values approach the vertical asymptotes, the tangent values increase or decrease without bound.

• No limits to how high or low the function outputs can go.
• The range remains unchanged, despite transformations like scaling or shifting.

Regardless of alterations to the function, such as stretching or translations, the range remains \( (-\infty, \infty) \), making the tangent function highly versatile.