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Chapter 1: Problem 123

Use a graphing utility to graph the function. $$ f(x)=2 \arcsin x $$

Short Answer

Expert verified
The graph of the function \(f(x) = 2\arcsin(x)\) lies within \(x\) values of -1 and 1 and \(y\) values of \(-\pi\) and \(\pi\), and has twice the vertical stretch compared to the basic \(\arcsin(x)\) function.

Step by step solution

01

Understand the basic shape of the arcsin function

The \(\arcsin(x)\) function typically looks like an 'S' lying on its side. It is defined only for \(x\) values from \(-1\) to \(1\) and its range lies between \(-\pi/2\) and \(\pi/2\).
02

Apply the vertical elongation

The function to be graphed is not just \(\arcsin(x)\), it's \(2\arcsin(x)\). A '2' multiplying a function has the effect of stretching the graph vertically by a factor of 2.
03

Use a graphing utility to create the graph.

At this stage, use a graphing utility or tool and graph \(y = 2\arcsin(x)\). Make sure it's set to the right mode (radians or degrees depending on what system you're most comfortable with).

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Key Concepts

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

Understanding the Arcsine Function
The arcsine function, denoted as \( \arcsin(x) \), is an inverse trigonometric function. It provides the angle whose sine is \( x \). This means if you know the sine value, \( \arcsin(x) \) gives you the angle for which the sine would be \( x \). The function is represented as an 'S' lying on its side, confined within a specific range.

Key points about the \( \arcsin(x) \) function:
  • It is defined only for the input values \( x \) between -1 and 1.
  • The output (or range) of this function is constrained within \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
  • The graph moves from the left to the right smoothly, transitioning between the angles.
The arcsine function is crucial in many applications, especially where solving for angles from given sine values is necessary. Its understanding forms the groundwork for comprehending more complex transformations, like vertical stretching.
Using a Graphing Utility
A graphing utility is a digital tool, either web-based or a standalone application, that enables the visual representation of mathematical functions. These utilities help in exploring the shape and structure of functions by plotting them on a coordinate grid.

For the function \( y = 2\arcsin(x) \), a graphing utility helps visualize how the arcsine function has been transformed. When using a graphing utility:
  • Ensure the function is entered correctly, making sure to use correct syntax.
  • Set the mode to either radians or degrees. Functions like \( \arcsin(x) \) are usually expressed in radians by default.
  • Identify how the vertical stretch impacts the graph as you compare it to the standard arcsine curve.
Whether on calculators, specialized software, or online graphing platforms, these tools bring clarity and precision, making it easier to analyze mathematical functions visually.
Exploring Vertical Stretch
Vertical stretch is a transformation that alters how "tall" a function appears on a graph. It affects the y-values of the function, effectively scaling them by a constant factor. When the coefficient in front of a function is greater than 1, it stretches the graph vertically.

For \( f(x) = 2\arcsin(x) \):
  • The coefficient 2 indicates a vertical stretch of the standard \( \arcsin(x) \) by a factor of 2.
  • This stretch doubles every y-value of the original arcsine graph, making the peaks and troughs twice as high or low.
  • It's essential to recognize that only the y-values are affected; the x-values remain within the same domain, \([-1, 1]\).
Vertical stretching doesn't alter the domain (the x-values), but it certainly changes how the function "looks" visually, making the peaks more pronounced. Understanding this helps in interpreting changes to functions across various transformations.

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Most popular questions from this chapter

Describe the behavior of \(f(\theta)=\sec \theta\) at the zeros of \(g(\theta)=\cos \theta\). Explain your reasoning.

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi)\), and sketch its graph.

In Exercises 109-112, sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \sin \theta=\frac{3}{4} $$

In Exercises 109-112, sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \cos \theta=\frac{5}{6} $$

In Exercises 19-34, use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arctan (-3) $$
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