Start with the parent graph y = tan(x)
Identify the period as pi
Mark vertical asymptotes at x = pi/2 + kpi, where k is any integer
Plot the x-intercepts at x = kpi, where k is any integer
Draw the repeating increasing curve between each pair of asymptotes
Make the curve pass through points like (-pi/4, -1), (0, 0), and (pi/4, 1)
Apply vertical shifts by adding or subtracting outside the tangent function
Apply horizontal shifts by changing x inside the function
Apply vertical stretches or compressions by multiplying the tangent output
Apply horizontal stretches or compressions by changing the coefficient of x inside the function
For y = a tan(b(x – h)) + k, use h for horizontal shift, k for vertical shift, a for vertical stretch, and pi/|b| for the period
Draw asymptotes using x = h + pi/(2|b|) + npi/|b|, where n is any integer
Plot the center point at (h, k)
Sketch each branch so it increases from negative infinity to positive infinity between asymptotes