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Chapter 4: Problem 83

Sketch a graph of the function. $$y=2 \arccos x$$

Short Answer

Expert verified
The graph of the function \(y=2 \arccos x\) is a vertically stretched version of the \(\arccos x\) function flipped on the line \(y=x\). It increases from \(0\) at \(x=1\) to \(2\pi\) at \(x=-1\), in the shape of an upward arch.

Step by step solution

01

Understanding the Arccosine Function

The arccosine function, \(\arccos x\), is the inverse function of the cosine function. It is defined for all \(x\) such that \(-1 \leq x \leq 1\). The range of the function is from \(0\) to \(\pi\). Thus, the range of \(2\arccos x\) is from \(0\) to \(2\pi\). When \(x=0\), \(2\arccos 0 = \pi\), and when \(x=1\), \(2\arccos 1 = 0\), when \(x=-1\), \(2\arccos(-1) = 2\pi\)
02

Plot the Key Points

Plotting the key points usually helps to begin the sketch. From the above step, the points to plot are at \((0, \pi)\), \((1, 0)\) and \((-1, 2\pi)\). The range of \(2\arccos x\) is from \(0\) to \(2\pi\). So, there are no outputs less than \(0\) or greater than \(2\pi\) for any inputs between \(-1\) and \(1\).
03

Understand the Symmetry

The graph of any inverse trigonometric function will always be symmetric in the line \(y=x\). However, as the function is \(2\arccos x\), the graph will be vertically stretched by a factor of 2. Thus, while sketching, remember to include the aspects of stretch.
04

Sketch the Graph

With the key points and the characteristics of the \(\arccos x\) function in mind, you can now sketch the graph. The function increases from \(0\) when \(x=1\) to \(2\pi\) when \(x=-1\), so the shape of the graph is coming from below in an arch shape going upwards from \(x=1\) to \(x=-1\). Because the graph of the arccosine function is a reduction of the cosine function, it will have a similar shape to the cosine function, but flipped on the line \(y=x\), and vertically stretched.

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