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Chapter 0: Problem 17

Sketch a graph of the function. $$f(x)=\tan 2 x$$

Short Answer

Expert verified
The graph of \( f(x) = \tan 2x \) is a tangent curve that repeats every \( \frac{\pi}{2} \) with asymptotes at \( x= \frac{\pi}{4} + n\frac{\pi}{2} \). The overall shape resembles the basic tangent function, but it's horizontally compressed.

Step by step solution

01

Identify the properties of the base function

The base function is \( \tan x \), which repeats every \( \pi \) and has asymptotes at \( x= \frac{\pi}{2} + n\pi \), where \( n \) is any integer.
02

Identify the transformation

The function \( f(x) = \tan 2x \) involves a horizontal compression by a factor of \( \frac{1}{2} \). This changes the period of \( \tan x \) from \( \pi \) to \( \frac{\pi}{2} \). The asymptotes also change to \( x= \frac{\pi}{4} + n\frac{\pi}{2} \).
03

Sketch the graph

Since there's no vertical or horizontal shift, the graph still passes through the origin. Plot the key points for one period from \( -\frac{\pi}{4} \) to \( \frac{\pi}{4} \) and then repeat the pattern. Draw vertical asymptotes at \( x= \frac{\pi}{4} + n\frac{\pi}{2} \) and sketch the function with these guidelines.

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Key Concepts

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

Tangent Function Properties
The tangent function, denoted by \( \tan x \), is one of the fundamental trigonometric functions. It plays a crucial role in various mathematical applications. Unlike the sine and cosine functions, which have distinctive wave-like patterns, the graph of \( \tan x \) displays a unique set of properties.
  • Periodicity: The tangent function is periodic with a period of \( \pi \). This means it repeats its values at intervals of \( \pi \).
  • Asymptotes: While the sine and cosine functions are bounded, the tangent function is unbounded and has vertical asymptotes. These occur wherever the denominator of the tangent function (i.e., the cosine function) equals zero, specifically at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
  • Odd Function: The function \( \tan x \) is an odd function, meaning it is symmetric with respect to the origin. Mathematically, this is represented as \( \tan(-x) = -\tan(x) \).
Understanding these properties is vital for effectively sketching and interpreting the graph of a tangent function.
Period and Asymptotes
Period and asymptotes are essential concepts for understanding and graphing trigonometric functions such as \( \tan x \).
  • Period of the function: The period of \( \tan x \) is the interval over which the function completes one full cycle before repeating itself. By default, \( \tan x \) has a period of \( \pi \). However, when a function such as \( \tan 2x \) is involved, the period is adjusted. For \( \tan kx \), the period becomes \( \frac{\pi}{k} \). Hence, for \( \tan 2x \), the period is \( \frac{\pi}{2} \).
  • Asymptotes of the function: Asymptotes for the tangent function occur where the function is undefined, resulting in vertical lines on the graph. For \( \tan kx \), the asymptotes are located at \( x = \frac{\pi}{2k} + n\frac{\pi}{k} \). For \( \tan 2x \), these are at \( x = \frac{\pi}{4} + n\frac{\pi}{2} \).
Understanding how the period and asymptotes modify with transformations allows for precise depictions of trigonometric graphs.
Horizontal Transformations
Horizontal transformations significantly impact the graph of a trigonometric function like \( \tan x \). They involve shifting or compressing the graph along the x-axis. When examining transformations, it is important to consider:
  • Horizontal Compression: This occurs when the period of the function decreases. For \( f(x) = \tan 2x \), the function experiences a horizontal compression. This is due to multiplying the angle by 2, which in turn halves the period to \( \frac{\pi}{2} \).
  • Horizontal Shift: Although not applicable to our specific function \( \tan 2x \) since there is no shift involved, a horizontal shift would involve adding a constant to the angle within the function. It moves the entire graph left or right.
Understanding how these transformations work helps in accurately sketching and predicting the behavior of trigonometric graphs and can alter periodicity and positioning of asymptotes.

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In general, if you have \(n\) chances of winning with a 1 -in- \(n\) chance on each try, the probability of winning at least once is \(1-\left(1-\frac{1}{n}\right)^{n} .\) As \(n\) gets larger, what number does this probability approach? (Hint: There is a very good reason that this question is in this section!)

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