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Chapter 6: Problem 46

Sketch the graph of the equation. $$y=\tan ^{-1} 2 x$$

Short Answer

Expert verified
The graph of \( y = \arctan(2x) \) is a horizontally compressed version of \( y = \arctan(x) \). It has horizontal asymptotes at \( y = \pm \frac{\pi}{2} \).

Step by step solution

01

Understand the Inverse Tangent Function

The function given is \( y = \arctan(2x) \). \( \arctan(x) \) is the inverse of the \( \tan(x) \) function, which means it returns an angle whose tangent is \( x \). The range of \( \arctan(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), and the domain is all real numbers. The graph is continuous, increasing, and approaches \( \frac{\pi}{2} \) and \( -\frac{\pi}{2} \) as \( x \to \infty \) and \( x \to -\infty \) respectively.
02

Identify the Effect of the Multiplier on x

The given function is \( y = \arctan(2x) \). This means \( x \) is scaled by 2. For the graph of \( y = \arctan(2x) \), this scaling horizontally compresses the graph towards the y-axis compared to \( y = \arctan(x) \). This means any features that occur for \( \arctan(x) \) will occur more quickly with respect to the x-axis.
03

Analyze Critical Points and Behavior

Identify some key points and behaviors for plotting: \( \arctan(0) = 0 \), so \( \arctan(2 \times 0) = 0 \). When \( x = 1 \), \( \arctan(2 \times 1) \approx \frac{\pi}{4} \). When \( x = -1 \), \( \arctan(2 \times (-1)) \approx -\frac{\pi}{4} \). As \( x \to \infty \), \( y \to \frac{\pi}{2} \) and as \( x \to -\infty \), \( y \to -\frac{\pi}{2} \).
04

Sketch the Graph

Begin graphing by marking \( (0,0) \) as a starting point. Indicate asymptotes at \( y = \pm \frac{\pi}{2} \). Plot a few points like \( (1, \frac{\pi}{4}) \) and \( (-1, -\frac{\pi}{4}) \). Draw a smooth curve through these points, showing the curve is increasing and approaches \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \) as asymptotic limits.

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Key Concepts

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

Graphing Functions
Graphing functions is a fundamental skill in understanding mathematical relationships. It allows us to visualize how one quantity depends on another by plotting the function on a coordinate grid. The function is typically represented as \( y = f(x) \) where each \( x \) value corresponds to a \( y \) value.

To graph any function:
  • Identify the type of function and specific characteristics such as symmetry, intercepts, and possible asymptotes.
  • Select a range of \( x \) values and calculate corresponding \( y \) values.
  • Plot these points on a graph and connect them to form a curve or line.
Understanding these principles will enable accurate plotting of even complex functions like inverse trigonometric functions.
Inverse Tangent Function
The inverse tangent function, often written as \( y = \arctan(x) \), is an essential concept in trigonometry. It is the inverse of the tangent function and returns the angle whose tangent is \( x \). This function has certain characteristics:
  • The range, or output, of \( \arctan(x) \) is limited to \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), making it different from the tangent function, which repeats every \( \pi \).
  • The domain, or input, includes all real numbers, providing a wide scope for input values.
  • The graph is continuous and steadily increasing, with horizontal asymptotes at \( y = -\frac{\pi}{2} \) as \( x \) approaches \( -\infty \), and \( y = \frac{\pi}{2} \) as \( x \) approaches \( +\infty \).
Recognizing the behavior of the inverse tangent is crucial for understanding how transformations, like scaling, affect its graph.
Function Transformation
Function transformations modify the appearance of a function's graph without changing its fundamental nature.

For the equation \( y = \arctan(2x) \), the transformation involves scaling the input, which compresses the graph horizontally.

Here's how transformations work:
  • A multiplier inside the function, such as \( 2x \), affects the x-values, causing horizontal stretching or compression. In this case, \( x \) is replaced by \( 2x \), compressing the graph towards the y-axis. This causes the graph to approach asymptotic limits more quickly.
  • Horizontal compression implies that all characteristics of the function occur sooner on the x-axis compared to the standard form.
By understanding these transformations, one can accurately predict and sketch the changes in the graph of a function.

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Most popular questions from this chapter

Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0$$ Simplify the resulting expression. $$\frac{x}{x^{2}-a^{2}}$$

Find the solutions of the equation that are in the interval \([\mathbf{0}, \mathbf{2} \pi)\). $$\cos u+\cos 2 u=0$$

Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0$$ Simplify the resulting expression. $$\frac{\sqrt{x^{2}-a^{2}}}{x}$$

If \(f(x)=\cos x,\) show that \(\frac{f(x+h)-f(x)}{h}=\cos x\left(\frac{\cos h-1}{h}\right)-\sin x\left(\frac{\sin h}{h}\right)\).

Graphically solve the trigonometric equation on the indicated interval to two decimal places. $$\tan \left(\frac{3}{2} x+\frac{1}{2}\right)=\frac{3}{2} \sin 2 x, \quad[-\pi, \pi]$$
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