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Chapter 4: Problem 92

Use a graphing utility to graph the function. $$f(x)=-3+\arctan (\pi x)$$

Short Answer

Expert verified
The graph of \(f(x) = -3 + \arctan (\pi x)\) is a horizontally compressed by a factor of 1/\pi and vertically translated downward by 3 units version of the standard \(\arctan x\) function graph. The horizontal asymptotes are at \(y=-\pi/2 -3\) and \(y= \pi/2 -3\). The function crosses the y-axis at \(y=-3\).

Step by step solution

01

Graphing the Basic Arctan Function

Start by familiarizing yourself with the graph of the basic \(\arctan x\) function without any modifications. This function will have a horizontal asymptote at \(y = -\pi/2\) and \(y = \pi/2\), and it will cross through the origin with a positive slope.
02

Apply the Multiplication Effect

Then, consider the effect of the multiplication of the variable \(x\) by \(\pi\) inside the \(\arctan\) function, resulting in the function \(\arctan (\pi x)\). This does not shift the graph left or right or up or down, but it does horizontally compress the graph by a factor of 1/\(\pi\). This means that every x-coordinate on the original function鈥檚 graph will be closer to the y-axis by a factor of 1/\(\pi\). Thus, the graph of \(\arctan (\pi x)\) will pass through the same horizontal asymptotes but will intersect the y-axis at \(\arctan(0) = 0\).
03

Apply the Translation Effect

Finally, consider the overall negative 3 that is added to the function \(-3 + \arctan (\pi x)\). This causes a downward shift of the entire graph by 3 units. As such, the horizontal asymptotes will also be moved downward by 3 units, resulting in asymptotes at \(y = -\pi/2 - 3\) and \(y = \pi/2 - 3\), and the function will intersect the y-axis at \(y = -3\).

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