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Chapter 5: Problem 82

Use a graphing utility to graph equation. $$y=\tan ^{-1}(x-1)$$

Short Answer

Expert verified
The graph of \(y = \tan^{-1}(x-1)\) appears as an 'S' curve with horizontal asymptotes at \(y = -\pi/2\) and \(y = \pi/2\).

Step by step solution

01

Initialize your Graphing Utility

First, open up your graphing utility. This can be an application, like Desmos or Geogebra, or a calculator that has a graphing feature. Make sure you are familiar with the basic operation of this tool.
02

Input the Function

Enter the provided function, \(y=\tan^{-1}(x-1)\), into your graphing utility. For many programs, this will mean typing 'atan(x-1)' into the function input space. Note that computers usually understand 'atan' to be the inverse tangent function, \(\tan^{-1}\). Make sure to also set a fitting viewing window.
03

Interpret the Graph

Observe and understand the graph made by your graphing utility. For the inverse tangent function, you will notice that it appears as an 'S' curve. The function approaches but never touches the horizontal asymptotes at \(y = -\pi/2\) and \(y = \pi/2\). This is characteristic of all inverse tangent functions, and seeing this shape potentially confirms the correctness of your input.

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Key Concepts

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

Graphing Utilities
Graphing utilities are powerful tools that help visualize mathematical functions. They are available as online platforms like Desmos and Geogebra, or as features on graphing calculators. By plotting equations, these tools make complex concepts more tangible.

Here's how to effectively use a graphing utility:
  • Familiarize yourself with the basic functions and controls. This will speed up the graphing process.
  • Input equations using the correct syntax. For inverse tangent functions, you might use 'atan' to represent \(\tan^{-1}\).
  • Adjust the viewing window to see the graph clearly. Make sure important features like asymptotes are visible.
By mastering these steps, you can fully utilize graphing utilities to explore and understand mathematical behaviors.
Inverse Tangent Function
The inverse tangent function, denoted as \(\tan^{-1}(x)\) or 'atan(x)', is a crucial concept in trigonometry. It is used to find an angle whose tangent is a given number. This function outputs angles in radians or degrees, typically between \(-\pi/2\) and \(\pi/2\).

When graphing \(y = \tan^{-1}(x-1)\), note that:
  • The function shifts one unit to the right due to the \(x-1\) modification. This type of transformation helps with understanding translations in graphs.
  • The graph takes on an 'S' shape, characteristic of inverse tangent functions.
  • This particular function will still have the range \(-\pi/2\) to \(\pi/2\), but horizontally adjusted due to the shift.
Understanding how shifts affect these functions can deepen your grasp on graph transformations.
Asymptotes
Asymptotes are lines that a graph approaches but never touches. They are common in many functions, including those involving inverse trigonometric functions.

For the inverse tangent function \(y = \tan^{-1}(x-1)\):
  • The horizontal asymptotes are at \(y = -\pi/2\) and \(y = \pi/2\). This shape is a signature of the inverse tangent graph.
  • These asymptotes indicate the end behavior of the function. The function value approaches these lines but remains bounded within them.
  • Understanding the role of asymptotes helps predict the long-term behavior of different functions, providing insights into their limits.
Grasping the concept of asymptotes aids in interpreting and explaining the extended nature of mathematical graphs.

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