Identifying the Graph of a Stretched Tangent

The graph has the shape of a tangent function.

• Step 1. One cycle extends from –4 to 4, so the period is $P=8.$ Since $P=\frac{\pi }{\left|B\right|},$ we have $B=\frac{\pi }{P}=\frac{\pi }{8}.$
• Step 2. The equation must have the form $f(x)=A\tan \left(\frac{\pi }{8}x\right).$
• Step 3. To find the vertical stretch $A,$ we can use the point $\left(2,2\right).$
  $$2=A\tan \left(\frac{\pi }{8}\cdot 2\right)=A\tan \left(\frac{\pi }{4}\right)$$

Because $\tan \left(\frac{\pi }{4}\right)=1,$ $A=2.$

This function would have a formula $f(x)=2\tan \left(\frac{\pi }{8}x\right).$