Question

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arctan x+\frac{\pi}{2} $$

Short answer

The graph of the function \(f(x) = \arctan x + \frac{\pi}{2}\) is the graph of the basic arctan function, but it is shifted upwards by \(\frac{\pi}{2}\) units. For any real number x, the value of the function lies in between \(0\) and \(\pi\) (exclusive), meaning the lines y=0 and y=\(\pi\) are horizontal asymptotes to this curve.

Step-by-step solution

Understand the Basic Graph of Arctangent

Understanding the base function which is \(y = \arctan x\) is the first step. The arctan function (also known as inverse tangent) graph is a curve that lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) (exclusive) for all real numbers x.

Apply the Transformation

The given function \(f(x) = \arctan x + \frac{\pi}{2}\) involves a vertical shift of the basic arctan graph. Adding a constant c to a function results into a shift upward by c units in the graph. So, the graph of this function will be the same as that of arctan x, but shifted up by \(\frac{\pi}{2}\) units.

Sketch the Graph

Sketch the basic arctan graph firstly. After that, move the entire graph upwards by \(\frac{\pi}{2}\) to draw the actual function graph. For any x, the function value (or y-coordinate) will be a point between \(0\) and \(\pi\) (exclusive). The graph will asymptote to these lines, not reaching them exactly.

Verify with a Graphing Utility

The last step is to verify this drawn graph with a graphing utility. Using a graphing tool like a graphical calculator or an online graphing tool, plot the function \(f(x) = \arctan x + \frac{\pi}{2}\). The sketch and the graph from the graphing utility should match.