## Trigonometry (10th Edition)

$\color{blue}{y=2 \tan{(\frac{\pi}{2})}}$
The graph looks like that of the basic tangent function $y=\tan{x}$. However, the period of the given function is $2\pi$, while that of the basic tangent function is $\pi$. RECALL: The period of the tangent function $y=\tan{(bx)}$ is $\frac{\pi}{b},b\gt 0$. Thus, $2\pi=\frac{\pi}{b} \\2b\pi=\pi \\\frac{2b\pi}{2\pi}=\frac{\pi}{2\pi} \\b=2$ Thus, the tentative equation of the function whose graph is given is $y=a\cdot \tan{(\frac{\pi}{2})}$. The given graph contains the point $(\frac{\pi}{2}, 2)$. Substituting the x and y values of this point into the tentative equation above gives: $y=a \cdot \tan{(\frac{\pi}{2})} \\2=a \cdot \tan{(\frac{\frac{\pi}{2}}{2})} \\2=a \cdot \tan{\frac{\pi}{4}} \\2=a \cdot 1 \\2=a$ Therefore, the equation of the the function whose graph is shown is $\color{blue}{y=2 \tan{(\frac{\pi}{2})}}$.