Graphing Variations of \({f( x) =\tan x}\) Using Transformations

Use the graph of \(f(x) = \tan x\) to graph \(g(x) =-\tan\! \left(\! x+\dfrac{\pi }{4}\right)\).

Solution Figure 78 illustrates the steps used in graphing \(g( x) =-\tan\! \left(\! x+\dfrac{\pi }{4} \right)\).

Begin by graphing \(f( x) =\tan x\). See Figure 78(a).

Replace the argument \(x\) by \(x+\dfrac{\pi }{4}\) to obtain \(y=\tan\! \left(\! x+\dfrac{\pi }{4} \right)\), which shifts the graph horizontally to the left \(\dfrac{\pi }{4}\) unit, as shown in Figure 78(b).

Multiply \(\tan\! \left(\! x+\dfrac{\pi }{4} \right)\) by \(-1\), which reflects the graph about the \(x\)-axis, and results in the graph of \(y =-\tan\! \left(\! x+\dfrac{\pi }{4} \right)\), as shown in Figure 78(c).