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Chapter 5: Problem 20

Find the period, and graph the function. $$y=\tan 4 x$$

Short Answer

Expert verified
The period of \( y = \tan 4x \) is \( \frac{\pi}{4} \).

Step by step solution

01

Identify the Basic Function

The basic function we're working with is the tangent function, which is given by \( y = \tan x \). The parent function \( y = \tan x \) has a period of \( \pi \).
02

Determine the Period of the Given Function

For a function of the form \( y = \tan(bx) \), the period is given by \( \frac{\pi}{|b|} \). In this case, \( b = 4 \), so the period of \( y = \tan 4x \) is \( \frac{\pi}{4} \).
03

Graph the Basic Tangent Function for One Period

To understand the transformation, sketch the graph of \( y = \tan x \) over a single period from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). The graph will have vertical asymptotes at these points and will pass through the origin \((0,0)\).
04

Apply the Period Transformation and Graph

Given the period \( \frac{\pi}{4} \), divide this interval into four equal parts to plot the graph of \( y = \tan 4x \). Start plotting from \( -\frac{\pi}{8} \) to \( \frac{\pi}{8} \), with vertical asymptotes at these boundaries, and draw the tangent curve through the origin. Extend this pattern to sketch the full graph over several periods.

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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, represented as \( y = \tan x \), is a fundamental trigonometric function. Unlike sine and cosine, which have a repeating pattern over \( 2\pi \), the tangent function has a period of \( \pi \). This means it repeats its values over every interval of \( \pi \) rather than \( 2\pi \).
  • The graph of \( y = \tan x \) features vertical asymptotes, which are lines the graph approaches but never touches or crosses.
  • These asymptotes occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer, and they result in the characteristic repeating pattern of the tangent graph.
  • The function is undefined at these points because the tangent of these angles results in division by zero.
The graph also goes through the origin, \( (0, 0) \), and it travels upwards to the right and downwards to the left, between its asymptotes.
Period of a Function
The period of a function is the distance over which the function's shape repeats. For trigonometric functions, understanding the period is crucial to graphing and analyzing behavior.
  • The standard period of the basic tangent function \( y = \tan x \) is \( \pi \).
  • When the function becomes \( y = \tan(bx) \), the period is adjusted by the factor \( b \). The new period can be found using the formula \( \frac{\pi}{|b|} \).
In the example \( y = \tan 4x \), the period is \( \frac{\pi}{4} \). This means that instead of repeating every \( \pi \), the function completes a full cycle every \( \frac{\pi}{4} \). This change indicates that the function will oscillate more frequently as it has a smaller interval over which it completes its behavior.
Graphing Trigonometric Functions
Graphing trigonometric functions like \( y = \tan 4x \) involves understanding how transformations affect the graph's shape and position. Here's a step-by-step guide on how to graph these functions:
  • Identify the basic function: Start with understanding the basic parent tangent function \( y = \tan x \) over one cycle, stretching from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
  • Calculate the new period: Using the period calculation, find \( \frac{\pi}{b} \) which transforms the graph. Here, it's \( \frac{\pi}{4} \).
  • Sketch the graph for the new period: Begin drawing the graph for one period from \( -\frac{\pi}{8} \) to \( \frac{\pi}{8} \), ensuring to place vertical asymptotes at these points.
  • Include key points: Plot the origin \((0,0)\) and observe the pattern forming.
  • Repeat the pattern: Extend the pattern to fill the desired graph length, repeating these steps over multiple periods.
This process helps you accurately reflect the behavior of the function on a graph, providing a visual representation of its oscillations and asymptotes.

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