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Chapter 5: Problem 12

Find the period, and graph the function. $$y=\frac{3}{4} \tan x$$

Short Answer

Expert verified
The period is \( \pi \), and the graph is a scaled tangent function.

Step by step solution

01

Understanding the Function

The given function is \( y = \frac{3}{4} \tan x \). This is a transformation of the basic tangent function \( \tan x \). The coefficient \( \frac{3}{4} \) affects the amplitude, which doesn't change the period of a tangent function.
02

Determine the Period of Tangent

The basic period of the tangent function \( \tan x \) is \( \pi \). The period of the function is calculated by dividing the period of the basic tangent function by the absolute value of the coefficient of \( x \). Since there is no coefficient affecting \( x \) here other than 1, the period remains \( \pi \).
03

Graph the Function

To graph \( y = \frac{3}{4} \tan x \), start by drawing the basic \( \tan x \) function, which has vertical asymptotes at \( x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \) etc. The function passes through the origin and repeats every \( \pi \) units. Multiply the output by \( \frac{3}{4} \) to scale it, making the graph less steep in comparison to \( y = \tan x \).
04

Identify Key Points and Vertical Asymptotes

At the key points between the asymptotes (e.g., \( x = 0, \pi, -\pi \)), the graph crosses through these points. The vertical asymptotes are unchanged from \( y = \tan x \), occurring at \(x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \), etc. To plot the graph accurately, identify a few specific points between the asymptotes and plot according to the scaling factor of \( \frac{3}{4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, denoted as \( \tan x \), plays a crucial role in trigonometry. It is defined as the ratio of the sine function to the cosine function:
  • \( \tan x = \frac{\sin x}{\cos x} \)
  • The tangent function is undefined when \( \cos x = 0 \), which is at odd multiples of \( \frac{\pi}{2} \), leading to vertical asymptotes.
  • This function repeats itself or is periodic every \( \pi \) units.
The graph of \( \tan x \) illustrates these characteristics, with waves that rise and fall within its period, looping infinitely. Unlike sine and cosine, which have bounds of \([-1, 1]\), the tangent function's range is all real numbers \((-\infty, \infty)\). This creates the distinct steep waves between each period.
Graphing
Graphing the tangent function involves recognizing its unique characteristics, which include vertical asymptotes and periodicity:
  • Identify key points: The graph always crosses the x-axis at \( x = 0, \pi, -\pi, \dots \)
  • Determine vertical asymptotes: These occur where \( \tan x \) is undefined, at \( x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \dots \)
  • The graph repeats every \( \pi \) units.
In the transformed function \( y = \frac{3}{4} \tan x \), the output values are scaled by \( \frac{3}{4} \). This affects the steepness of the graph, but not its asymptotes or period. For example, the tangent waveform becomes less steep, but it will still extend from top to bottom between the asymptotes. Understanding these principles helps in sketching or interpreting the function's graph properly.
Periodicity
Periodicity refers to how a function repeats itself over intervals. For the tangent function, this occurs every \( \pi \) units. This means that if you move \( \pi \) units along the x-axis, the function's values repeat themselves identically.
  • The period of \( \tan x \) is \( \pi \), regardless of amplitude.
  • Transformation does not change its core period, unless the internal workings (i.e., a coefficient multiplying \( x \)) are altered.
For the given exercise, since the function is \( \frac{3}{4} \tan x \), the period remains \( \pi \). This stable periodicity makes tangent functions predictable and largely unchanged by amplitude variations. Being aware of a function's period is vital, as it assists in plotting, analyzing, and understanding the function's behavior over its domain.

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Most popular questions from this chapter

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. $$\sec t, \tan t ; \quad \text { Quadrant II }$$

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