1. The problem is to understand how to graph tangent functions. 2. The tangent function is defined as $y=\tan(x)$, which is the ratio of sine to cosine: $\tan(x) = \frac{\sin(x)}{\cos(x)}$. 3. Important rules for graphing tangent: - The function has vertical asymptotes where $\cos(x) = 0$, i.e., at $x=\frac{\pi}{2} + k\pi$ for any integer $k$. - The function is periodic with period $\pi$. - The function passes through the origin $(0,0)$. 4. To graph $y=\tan(x)$: - Plot points where tangent is zero: $x=0, \pm \pi, \pm 2\pi, ...$ - Draw vertical dashed lines (asymptotes) at $x=\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, ...$ - Sketch the curve increasing from $-\infty$ to $+\infty$ between asymptotes. 5. The graph repeats every $\pi$ units, so the pattern between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ repeats across the x-axis. Final answer: The tangent function graph has vertical asymptotes at $x=\frac{\pi}{2}+k\pi$, zeros at $x=k\pi$, and repeats every $\pi$ units with the curve increasing from negative to positive infinity between asymptotes.